TAO: Optimization Solvers

The Toolkit for Advanced Optimization (TAO) focuses on algorithms for the solution of large-scale optimization problems on high-performance architectures. Methods are available for

• Nonlinear Least-Squares

• Quadratic Solvers

• Unconstrained Minimization

• Bound-Constrained Optimization

• Generally Constrained Solvers

• Complementarity

• PDE-constrained Optimization

Getting Started: A Simple TAO Example

To help the user start using TAO immediately, we introduce here a simple uniprocessor example. Please read TAO Algorithms for a more in-depth discussion on using the TAO solvers. The code presented below minimizes the extended Rosenbrock function \(f: \mathbb R^n \to \mathbb R\) defined by

\[f(x) = \sum_{i=0}^{m-1} \left( \alpha(x_{2i+1}-x_{2i}^2)^2 + (1-x_{2i})^2 \right),\]

where \(n = 2m\) is the number of variables. Note that while we use the C language to introduce the TAO software, the package is fully usable from C++ and Fortran. PETSc for Fortran Users discusses additional issues concerning Fortran usage.

The code in the example contains many of the components needed to write most TAO programs and thus is illustrative of the features present in complex optimization problems. Note that for display purposes we have omitted some nonessential lines of code as well as the (essential) code required for the routine FormFunctionGradient, which evaluates the function and gradient, and the code for FormHessian, which evaluates the Hessian matrix for Rosenbrock’s function. The complete code is available in $TAO_DIR/src/unconstrained/tutorials/rosenbrock1.c. The following sections annotate the lines of code in the example.

Listing: src/tao/unconstrained/tutorials/rosenbrock1.c

#include <petsctao.h>
typedef struct {
  PetscInt  n;     /* dimension */
  PetscReal alpha; /* condition parameter */
  PetscBool chained;
} AppCtx;

/* -------------- User-defined routines ---------- */
PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal *, Vec, void *);
PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void *);

int main(int argc, char **argv)
{
  PetscReal   zero = 0.0;
  Vec         x; /* solution vector */
  Mat         H;
  Tao         tao; /* Tao solver context */
  PetscBool   flg, test_lmvm = PETSC_FALSE;
  PetscMPIInt size; /* number of processes running */
  AppCtx      user; /* user-defined application context */
  KSP         ksp;
  PC          pc;
  Mat         M;
  Vec         in, out, out2;
  PetscReal   mult_solve_dist;

  /* Initialize TAO and PETSc */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
  PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
  PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Incorrect number of processors");

  /* Initialize problem parameters */
  user.n       = 2;
  user.alpha   = 99.0;
  user.chained = PETSC_FALSE;
  /* Check for command line arguments to override defaults */
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-n", &user.n, &flg));
  PetscCall(PetscOptionsGetReal(NULL, NULL, "-alpha", &user.alpha, &flg));
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-chained", &user.chained, &flg));
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-test_lmvm", &test_lmvm, &flg));

  /* Allocate vectors for the solution and gradient */
  PetscCall(VecCreateSeq(PETSC_COMM_SELF, user.n, &x));
  PetscCall(MatCreateSeqBAIJ(PETSC_COMM_SELF, 2, user.n, user.n, 1, NULL, &H));

  /* The TAO code begins here */

  /* Create TAO solver with desired solution method */
  PetscCall(TaoCreate(PETSC_COMM_SELF, &tao));
  PetscCall(TaoSetType(tao, TAOLMVM));

  /* Set solution vec and an initial guess */
  PetscCall(VecSet(x, zero));
  PetscCall(TaoSetSolution(tao, x));

  /* Set routines for function, gradient, hessian evaluation */
  PetscCall(TaoSetObjectiveAndGradient(tao, NULL, FormFunctionGradient, &user));
  PetscCall(TaoSetHessian(tao, H, H, FormHessian, &user));

  /* Test the LMVM matrix */
  if (test_lmvm) PetscCall(PetscOptionsSetValue(NULL, "-tao_type", "bqnktr"));

  /* Check for TAO command line options */
  PetscCall(TaoSetFromOptions(tao));

  /* SOLVE THE APPLICATION */
  PetscCall(TaoSolve(tao));

  /* Test the LMVM matrix */
  if (test_lmvm) {
    PetscCall(TaoGetKSP(tao, &ksp));
    PetscCall(KSPGetPC(ksp, &pc));
    PetscCall(PCLMVMGetMatLMVM(pc, &M));
    PetscCall(VecDuplicate(x, &in));
    PetscCall(VecDuplicate(x, &out));
    PetscCall(VecDuplicate(x, &out2));
    PetscCall(VecSet(in, 1.0));
    PetscCall(MatMult(M, in, out));
    PetscCall(MatSolve(M, out, out2));
    PetscCall(VecAXPY(out2, -1.0, in));
    PetscCall(VecNorm(out2, NORM_2, &mult_solve_dist));
    if (mult_solve_dist < 1.e-11) {
      PetscCall(PetscPrintf(PetscObjectComm((PetscObject)tao), "error between LMVM MatMult and MatSolve: < 1.e-11\n"));
    } else if (mult_solve_dist < 1.e-6) {
      PetscCall(PetscPrintf(PetscObjectComm((PetscObject)tao), "error between LMVM MatMult and MatSolve: < 1.e-6\n"));
    } else {
      PetscCall(PetscPrintf(PetscObjectComm((PetscObject)tao), "error between LMVM MatMult and MatSolve: %e\n", (double)mult_solve_dist));
    }
    PetscCall(VecDestroy(&in));
    PetscCall(VecDestroy(&out));
    PetscCall(VecDestroy(&out2));
  }

  PetscCall(TaoDestroy(&tao));
  PetscCall(VecDestroy(&x));
  PetscCall(MatDestroy(&H));

  PetscCall(PetscFinalize());
return ierr;}

TAO Workflow

Many TAO applications will follow an ordered set of procedures for solving an optimization problem: The user creates a Tao context and selects a default algorithm. Call-back routines as well as vector (Vec) and matrix (Mat) data structures are then set. These call-back routines will be used for evaluating the objective function, gradient, and perhaps the Hessian matrix. The user then invokes TAO to solve the optimization problem and finally destroys the Tao context. A list of the necessary functions for performing these steps using TAO is shown below.

TaoCreate(MPI_Comm comm, Tao *tao);
TaoSetType(Tao tao, TaoType type);
TaoSetSolution(Tao tao, Vec x);
TaoSetObjectiveAndGradient(Tao tao, Vec g, PetscErrorCode (*FormFGradient)(Tao, Vec, PetscReal*, Vec, void*), void *user);
TaoSetHessian(Tao tao, Mat H, Mat Hpre, PetscErrorCode (*FormHessian)(Tao, Vec, Mat, Mat, void*), void *user);
TaoSolve(Tao tao);
TaoDestroy(Tao tao);

Note that the solver algorithm selected through the function TaoSetType() can be overridden at runtime by using an options database. Through this database, the user not only can select a minimization method (e.g., limited-memory variable metric, conjugate gradient, Newton with line search or trust region) but also can prescribe the convergence tolerance, set various monitoring routines, set iterative methods and preconditions for solving the linear systems, and so forth. See TAO Algorithms for more information on the solver methods available in TAO.

Header File

TAO applications written in C/C++ should have the statement

#include <petsctao.h>

in each file that uses a routine in the TAO libraries.

Creation and Destruction

A TAO solver can be created by calling the

TaoCreate(MPI_Comm, Tao*);

routine. Much like creating PETSc vector and matrix objects, the first argument is an MPI communicator. An MPI 1 communicator indicates a collection of processors that will be used to evaluate the objective function, compute constraints, and provide derivative information. When only one processor is being used, the communicator PETSC_COMM_SELF can be used with no understanding of MPI. Even parallel users need to be familiar with only the basic concepts of message passing and distributed-memory computing. Most applications running TAO in parallel environments can employ the communicator PETSC_COMM_WORLD to indicate all processes known to PETSc in a given run.

The routine

TaoSetType(Tao, TaoType);

can be used to set the algorithm TAO uses to solve the application. The various types of TAO solvers and the flags that identify them will be discussed in the following sections. The solution method should be carefully chosen depending on the problem being solved. Some solvers, for instance, are meant for problems with no constraints, whereas other solvers acknowledge constraints in the problem and handle them accordingly. The user must also be aware of the derivative information that is available. Some solvers require second-order information, while other solvers require only gradient or function information. The command line option -tao_type followed by a TAO method will override any method specified by the second argument. The command line option -tao_type bqnls, for instance, will specify the limited-memory quasi-Newton line search method for bound-constrained problems. Note that the TaoType variable is a string that requires quotation marks in an application program, but quotation marks are not required at the command line.

Each TAO solver that has been created should also be destroyed by using the

TaoDestroy(Tao tao);

command. This routine frees the internal data structures used by the solver.

Command-line Options

Additional options for the TAO solver can be be set from the command line by using the

TaoSetFromOptions(Tao)

routine. This command also provides information about runtime options when the user includes the -help option on the command line.

In addition to common command line options shared by all TAO solvers, each TAO method also implements its own specialized options. Please refer to the documentation for individual methods for more details.

Defining Variables

In all the optimization solvers, the application must provide a Vec object of appropriate dimension to represent the variables. This vector will be cloned by the solvers to create additional work space within the solver. If this vector is distributed over multiple processors, it should have a parallel distribution that allows for efficient scaling, inner products, and function evaluations. This vector can be passed to the application object by using the

TaoSetSolution(Tao, Vec);

routine. When using this routine, the application should initialize the vector with an approximate solution of the optimization problem before calling the TAO solver. This vector will be used by the TAO solver to store the solution. Elsewhere in the application, this solution vector can be retrieved from the application object by using the

TaoGetSolution(Tao, Vec*);

routine. This routine takes the address of a Vec in the second argument and sets it to the solution vector used in the application.

User Defined Call-back Routines

Users of TAO are required to provide routines that perform function evaluations. Depending on the solver chosen, they may also have to write routines that evaluate the gradient vector and Hessian matrix.

Application Context

Writing a TAO application may require use of an application context. An application context is a structure or object defined by an application developer, passed into a routine also written by the application developer, and used within the routine to perform its stated task.

For example, a routine that evaluates an objective function may need parameters, work vectors, and other information. This information, which may be specific to an application and necessary to evaluate the objective, can be collected in a single structure and used as one of the arguments in the routine. The address of this structure will be cast as type (void*) and passed to the routine in the final argument. Many examples of these structures are included in the TAO distribution.

This technique offers several advantages. In particular, it allows for a uniform interface between TAO and the applications. The fundamental information needed by TAO appears in the arguments of the routine, while data specific to an application and its implementation is confined to an opaque pointer. The routines can access information created outside the local scope without the use of global variables. The TAO solvers and application objects will never access this structure, so the application developer has complete freedom to define it. If no such structure or needed by the application then a NULL pointer can be used.

Objective Function and Gradient Routines

TAO solvers that minimize an objective function require the application to evaluate the objective function. Some solvers may also require the application to evaluate derivatives of the objective function. Routines that perform these computations must be identified to the application object and must follow a strict calling sequence.

Routines should follow the form

PetscErrorCode EvaluateObjective(Tao, Vec, PetscReal*, void*);

in order to evaluate an objective function \(f: \, \mathbb R^n \to \mathbb R\). The first argument is the TAO Solver object, the second argument is the \(n\)-dimensional vector that identifies where the objective should be evaluated, and the fourth argument is an application context. This routine should use the third argument to return the objective value evaluated at the point specified by the vector in the second argument.

This routine, and the application context, should be passed to the application object by using the

TaoSetObjective(Tao, PetscErrorCode(*)(Tao,Vec,PetscReal*,void*), void*);

routine. The first argument in this routine is the TAO solver object, the second argument is a function pointer to the routine that evaluates the objective, and the third argument is the pointer to an appropriate application context. Although the final argument may point to anything, it must be cast as a (void*) type. This pointer will be passed back to the developer in the fourth argument of the routine that evaluates the objective. In this routine, the pointer can be cast back to the appropriate type. Examples of these structures and their usage are provided in the distribution.

Many TAO solvers also require gradient information from the application The gradient of the objective function is specified in a similar manner. Routines that evaluate the gradient should have the calling sequence

PetscErrorCode EvaluateGradient(Tao, Vec, Vec, void*);

where the first argument is the TAO solver object, the second argument is the variable vector, the third argument is the gradient vector, and the fourth argument is the user-defined application context. Only the third argument in this routine is different from the arguments in the routine for evaluating the objective function. The numbers in the gradient vector have no meaning when passed into this routine, but they should represent the gradient of the objective at the specified point at the end of the routine. This routine, and the user-defined pointer, can be passed to the application object by using the

TaoSetGradient(Tao, Vec, PetscErrorCode (*)(Tao,Vec,Vec,void*), void*);

routine. In this routine, the first argument is the Tao object, the second argument is the optional vector to hold the computed gradient, the third argument is the function pointer, and the fourth object is the application context, cast to (void*).

Instead of evaluating the objective and its gradient in separate routines, TAO also allows the user to evaluate the function and the gradient in the same routine. In fact, some solvers are more efficient when both function and gradient information can be computed in the same routine. These routines should follow the form

PetscErrorCode EvaluateFunctionAndGradient(Tao, Vec, PetscReal*, Vec, void*);

where the first argument is the TAO solver and the second argument points to the input vector for use in evaluating the function and gradient. The third argument should return the function value, while the fourth argument should return the gradient vector. The fifth argument is a pointer to a user-defined context. This context and the name of the routine should be set with the call

TaoSetObjectiveAndGradient(Tao, Vec PetscErrorCode (*)(Tao,Vec,PetscReal*,Vec,void*), void*);

where the arguments are the TAO application, the optional vector to be used to hold the computed gradient, a function pointer, and a pointer to a user-defined context.

The TAO example problems demonstrate the use of these application contexts as well as specific instances of function, gradient, and Hessian evaluation routines. All these routines should return the integer \(0\) after successful completion and a nonzero integer if the function is undefined at that point or an error occurred.

Hessian Evaluation

Some optimization routines also require a Hessian matrix from the user. The routine that evaluates the Hessian should have the form

PetscErrorCode EvaluateHessian(Tao, Vec, Mat, Mat, void*);

where the first argument of this routine is a TAO solver object. The second argument is the point at which the Hessian should be evaluated. The third argument is the Hessian matrix, and the sixth argument is a user-defined context. Since the Hessian matrix is usually used in solving a system of linear equations, a preconditioner for the matrix is often needed. The fourth argument is the matrix that will be used for preconditioning the linear system; in most cases, this matrix will be the same as the Hessian matrix. The fifth argument is the flag used to set the Hessian matrix and linear solver in the routine KSPSetOperators().

One can set the Hessian evaluation routine by calling the

TaoSetHessian(Tao, Mat, Mat, PetscErrorCode (*)(Tao,Vec,Mat,Mat,void*), void*);

routine. The first argument is the TAO Solver object. The second and third arguments are, respectively, the Mat object where the Hessian will be stored and the Mat object that will be used for the preconditioning (they may be the same). The fourth argument is the function that evaluates the Hessian, and the fifth argument is a pointer to a user-defined context, cast to (void*).

Finite Differences

Finite-difference approximations can be used to compute the gradient and the Hessian of an objective function. These approximations will slow the solve considerably and are recommended primarily for checking the accuracy of hand-coded gradients and Hessians. These routines are

TaoDefaultComputeGradient(Tao, Vec, Vec, void*);

and

TaoDefaultComputeHessian(Tao, Vec, Mat*, Mat*,void*);

respectively. They can be set by using TaoSetGradient() and TaoSetHessian() or through the options database with the options -tao_fdgrad and -tao_fd, respectively.

The efficiency of the finite-difference Hessian can be improved if the coloring of the matrix is known. If the application programmer creates a PETSc MatFDColoring object, it can be applied to the finite-difference approximation by setting the Hessian evaluation routine to

TaoDefaultComputeHessianColor(Tao, Vec, Mat*, Mat*, void*);

and using the MatFDColoring object as the last (void *) argument to TaoSetHessian().

One also can use finite-difference approximations to directly check the correctness of the gradient and/or Hessian evaluation routines. This process can be initiated from the command line by using the special TAO solver tao_fd_test together with the option -tao_test_gradient or -tao_test_hessian.

Matrix-Free Methods

TAO fully supports matrix-free methods. The matrices specified in the Hessian evaluation routine need not be conventional matrices; instead, they can point to the data required to implement a particular matrix-free method. The matrix-free variant is allowed only when the linear systems are solved by an iterative method in combination with no preconditioning (PCNONE or -pc_type none), a user-provided preconditioner matrix, or a user-provided preconditioner shell (PCSHELL). In other words, matrix-free methods cannot be used if a direct solver is to be employed. Details about using matrix-free methods are provided in the User-Guide.

Fig. 7 Tao use of PETSc and callbacks

Constraints

Some optimization problems also impose constraints on the variables or intermediate application states. The user defines these constraints through the appropriate TAO interface functions and call-back routines where necessary.

Variable Bounds

The simplest type of constraint on an optimization problem puts lower or upper bounds on the variables. Vectors that represent lower and upper bounds for each variable can be set with the

TaoSetVariableBounds(Tao, Vec, Vec);

command. The first vector and second vector should contain the lower and upper bounds, respectively. When no upper or lower bound exists for a variable, the bound may be set to PETSC_INFINITY or PETSC_NINFINITY. After the two bound vectors have been set, they may be accessed with the command TaoGetVariableBounds().

Since not all solvers recognize the presence of bound constraints on variables, the user must be careful to select a solver that acknowledges these bounds.

General Constraints

Some TAO algorithms also support general constraints as a linear or nonlinear function of the optimization variables. These constraints can be imposed either as equalities or inequalities. TAO currently does not make any distinctions between linear and nonlinear constraints, and implements them through the same software interfaces.

In the equality constrained case, TAO assumes that the constraints are formulated as \(c_e(x) = 0\) and requires the user to implement a call-back routine for evaluating \(c_e(x)\) at a given vector of optimization variables,

PetscErrorCode EvaluateEqualityConstraints(Tao, Vec, Vec, void*);

As in the previous call-back routines, the first argument is the TAO solver object. The second and third arguments are the vector of optimization variables (input) and vector of equality constraints (output), respectively. The final argument is a pointer to the user-defined application context, cast into (void*).

Generally constrained TAO algorithms also require a second user call-back function to compute the constraint Jacobian matrix \(\nabla_x c_e(x)\),

PetscErrorCode EvaluateEqualityJacobian(Tao, Vec, Mat, Mat, void*);

where the first and last arguments are the TAO solver object and the application context pointer as before. The second argument is the vector of optimization variables at which the computation takes place. The third and fourth arguments are the constraint Jacobian and its pseudo-inverse (optional), respectively. The pseudoinverse is optional, and if not available, the user can simply set it to the constraint Jacobian itself.

These call-back functions are then given to the TAO solver using the interface functions

TaoSetEqualityConstraintsRoutine(Tao, Vec, PetscErrorCode (*)(Tao,Vec,Vec,void*), void*);

and

TaoSetJacobianEqualityRoutine(Tao, Mat, Mat, PetscErrorCode (*)(Tao,Vec,Mat,Mat,void*), void*);

Inequality constraints are assumed to be formulated as \(c_i(x) \leq 0\) and follow the same workflow as equality constraints using the TaoSetInequalityConstraintsRoutine() and TaoSetJacobianInequalityRoutine() interfaces.

Some TAO algorithms may adopt an alternative double-sided \(c_l \leq c_i(x) \leq c_u\) formulation and require the lower and upper bounds \(c_l\) and \(c_u\) to be set using the TaoSetInequalityBounds(Tao,Vec,Vec) interface. Please refer to the documentation for each TAO algorithm for further details.

Solving

Once the application and solver have been set up, the solver can be

TaoSolve(Tao);

routine. We discuss several universal options below.

Convergence

Although TAO and its solvers set default parameters that are useful for many problems, the user may need to modify these parameters in order to change the behavior and convergence of various algorithms.

One convergence criterion for most algorithms concerns the number of digits of accuracy needed in the solution. In particular, the convergence test employed by TAO attempts to stop when the error in the constraints is less than \(\epsilon_{crtol}\) and either

\[\begin{array}{lcl} ||g(X)|| &\leq& \epsilon_{gatol}, \\ ||g(X)||/|f(X)| &\leq& \epsilon_{grtol}, \quad \text{or} \\ ||g(X)||/|g(X_0)| &\leq& \epsilon_{gttol}, \end{array}\]

where \(X\) is the current approximation to the true solution \(X^*\) and \(X_0\) is the initial guess. \(X^*\) is unknown, so TAO estimates \(f(X) - f(X^*)\) with either the square of the norm of the gradient or the duality gap. A relative tolerance of \(\epsilon_{frtol}=0.01\) indicates that two significant digits are desired in the objective function. Each solver sets its own convergence tolerances, but they can be changed by using the routine TaoSetTolerances(). Another set of convergence tolerances terminates the solver when the norm of the gradient function (or Lagrangian function for bound-constrained problems) is sufficiently close to zero.

Other stopping criteria include a minimum trust-region radius or a maximum number of iterations. These parameters can be set with the routines TaoSetTrustRegionTolerance() and TaoSetMaximumIterations() Similarly, a maximum number of function evaluations can be set with the command TaoSetMaximumFunctionEvaluations(). -tao_max_it, and -tao_max_funcs.

Viewing Status

To see parameters and performance statistics for the solver, the routine

TaoView(Tao tao)

can be used. This routine will display to standard output the number of function evaluations need by the solver and other information specific to the solver. This same output can be produced by using the command line option -tao_view.

The progress of the optimization solver can be monitored with the runtime option -tao_monitor. Although monitoring routines can be customized, the default monitoring routine will print out several relevant statistics to the screen.

The user also has access to information about the current solution. The current iteration number, objective function value, gradient norm, infeasibility norm, and step length can be retrieved with the following command.

TaoGetSolutionStatus(Tao tao, PetscInt* iterate, PetscReal* f,
                  PetscReal* gnorm, PetscReal* cnorm, PetscReal* xdiff,
                  TaoConvergedReason* reason)

The last argument returns a code that indicates the reason that the solver terminated. Positive numbers indicate that a solution has been found, while negative numbers indicate a failure. A list of reasons can be found in the manual page for TaoGetConvergedReason().

Obtaining a Solution

After exiting the TaoSolve() function, the solution and the gradient can be recovered with the following routines.

TaoGetSolution(Tao, Vec*);
TaoGetGradient(Tao, Vec*, NULL, NULL);

Note that the Vec returned by TaoGetSolution() will be the same vector passed to TaoSetSolution(). This information can be obtained during user-defined routines such as a function evaluation and customized monitoring routine or after the solver has terminated.

Special Problem structures

Certain special classes of problems solved with TAO utilize specialized code interfaces that are described below per problem type.

PDE-constrained Optimization

TAO solves PDE-constrained optimization problems of the form

\[\begin{array}{ll} \displaystyle \min_{u,v} & f(u,v) \\ \text{subject to} & g(u,v) = 0, \end{array}\]

where the state variable \(u\) is the solution to the discretized partial differential equation defined by \(g\) and parametrized by the design variable \(v\), and \(f\) is an objective function. The Lagrange multipliers on the constraint are denoted by \(y\). This method is set by using the linearly constrained augmented Lagrangian TAO solver tao_lcl.

We make two main assumptions when solving these problems: the objective function and PDE constraints have been discretized so that we can treat the optimization problem as finite dimensional and \(\nabla_u g(u,v)\) is invertible for all \(u\) and \(v\).

Unlike other TAO solvers where the solution vector contains only the optimization variables, PDE-constrained problems solved with tao_lcl combine the design and state variables together in a monolithic solution vector \(x^T = [u^T, v^T]\). Consequently, the user must provide index sets to separate the two,

TaoSetStateDesignIS(Tao, IS, IS);

where the first IS is a PETSc IndexSet containing the indices of the state variables and the second IS the design variables.

PDE constraints have the general form \(g(x) = 0\), where \(c: \mathbb R^n \to \mathbb R^m\). These constraints should be specified in a routine, written by the user, that evaluates \(g(x)\). The routine that evaluates the constraint equations should have the form

PetscErrorCode EvaluateConstraints(Tao, Vec, Vec, void*);

The first argument of this routine is a TAO solver object. The second argument is the variable vector at which the constraint function should be evaluated. The third argument is the vector of function values \(g(x)\), and the fourth argument is a pointer to a user-defined context. This routine and the user-defined context should be set in the TAO solver with the

TaoSetConstraintsRoutine(Tao, Vec, PetscErrorCode (*)(Tao,Vec,Vec,void*), void*);

command. In this function, the first argument is the TAO solver object, the second argument a vector in which to store the constraints, the third argument is a function point to the routine for evaluating the constraints, and the fourth argument is a pointer to a user-defined context.

The Jacobian of \(g(x)\) is the matrix in \(\mathbb R^{m \times n}\) such that each column contains the partial derivatives of \(g(x)\) with respect to one variable. The evaluation of the Jacobian of \(g\) should be performed by calling the

PetscErrorCode JacobianState(Tao, Vec, Mat, Mat, Mat, void*);
PetscErrorCode JacobianDesign(Tao, Vec, Mat*, void*);

routines. In these functions, The first argument is the TAO solver object. The second argument is the variable vector at which to evaluate the Jacobian matrix, the third argument is the Jacobian matrix, and the last argument is a pointer to a user-defined context. The fourth and fifth arguments of the Jacobian evaluation with respect to the state variables are for providing PETSc matrix objects for the preconditioner and for applying the inverse of the state Jacobian, respectively. This inverse matrix may be PETSC_NULL, in which case TAO will use a PETSc Krylov subspace solver to solve the state system. These evaluation routines should be registered with TAO by using the

TaoSetJacobianStateRoutine(Tao, Mat, Mat, Mat,
                        PetscErrorCode (*)(Tao,Vec,Mat,Mat,void*),
                        void*);
TaoSetJacobianDesignRoutine(Tao, Mat,
                        PetscErrorCode (*)(Tao,Vec,Mat*,void*),
                        void*);

routines. The first argument is the TAO solver object, and the second argument is the matrix in which the Jacobian information can be stored. For the state Jacobian, the third argument is the matrix that will be used for preconditioning, and the fourth argument is an optional matrix for the inverse of the state Jacobian. One can use PETSC_NULL for this inverse argument and let PETSc apply the inverse using a KSP method, but faster results may be obtained by manipulating the structure of the Jacobian and providing an inverse. The fifth argument is the function pointer, and the sixth argument is an optional user-defined context. Since no solve is performed with the design Jacobian, there is no need to provide preconditioner or inverse matrices.

Nonlinear Least Squares

For nonlinear least squares applications, we are solving the optimization problem

\[\min_{x} \;\frac{1}{2}||r(x)||_2^2. \]

For these problems, the objective function value should be computed as a vector of residuals, \(r(x)\), computed with a function of the form

PetscErrorCode EvaluateResidual(Tao, Vec, Vec, void*);

and set with the

TaoSetResidualRoutine(Tao, PetscErrorCode (*)(Tao,Vec,Vec,void*), void*);

routine. If required by the algorithm, the Jacobian of the residual, \(J = \partial r(x) / \partial x\), should be computed with a function of the form

PetscErrorCode EvaluateJacobian(Tao, Vec, Mat, void*);

and set with the

TaoSetJacobianResidualRoutine(Tao, PetscErrorCode (*)(Tao,Vec,Mat,void*), void *);

routine.

Complementarity

Complementarity applications have equality constraints in the form of nonlinear equations \(C(X) = 0\), where \(C: \mathbb R^n \to \mathbb R^m\). These constraints should be specified in a routine written by the user with the form

PetscErrorCode EqualityConstraints(Tao, Vec, Vec, void*);

that evaluates \(C(X)\). The first argument of this routine is a TAO Solver object. The second argument is the variable vector \(X\) at which the constraint function should be evaluated. The third argument is the output vector of function values \(C(X)\), and the fourth argument is a pointer to a user-defined context.

This routine and the user-defined context must be registered with TAO by using the

TaoSetConstraintRoutine(Tao, Vec, PetscErrorCode (*)(Tao,Vec,Vec,void*), void*);

command. In this command, the first argument is TAO Solver object, the second argument is vector in which to store the function values, the third argument is the user-defined routine that evaluates \(C(X)\), and the fourth argument is a pointer to a user-defined context that will be passed back to the user.

The Jacobian of the function is the matrix in \(\mathbb R^{m \times n}\) such that each column contains the partial derivatives of \(f\) with respect to one variable. The evaluation of the Jacobian of \(C\) should be performed in a routine of the form

PetscErrorCode EvaluateJacobian(Tao, Vec, Mat, Mat, void*);

In this function, the first argument is the TAO Solver object and the second argument is the variable vector at which to evaluate the Jacobian matrix. The third argument is the Jacobian matrix, and the sixth argument is a pointer to a user-defined context. Since the Jacobian matrix may be used in solving a system of linear equations, a preconditioner for the matrix may be needed. The fourth argument is the matrix that will be used for preconditioning the linear system; in most cases, this matrix will be the same as the Hessian matrix. The fifth argument is the flag used to set the Jacobian matrix and linear solver in the routine KSPSetOperators().

This routine should be specified to TAO by using the

TaoSetJacobianRoutine(Tao, Mat, Mat, PetscErrorCode (*)(Tao,Vec,Mat,Mat,void*), void*);

command. The first argument is the TAO Solver object; the second and third arguments are the Mat objects in which the Jacobian will be stored and the Mat object that will be used for the preconditioning (they may be the same), respectively. The fourth argument is the function pointer; and the fifth argument is an optional user-defined context. The Jacobian matrix should be created in a way such that the product of it and the variable vector can be stored in the constraint vector.

TAO Algorithms

TAO includes a variety of optimization algorithms for several classes of problems (unconstrained, bound-constrained, and PDE-constrained minimization, nonlinear least-squares, and complementarity). The TAO algorithms for solving these problems are detailed in this section, a particular algorithm can chosen by using the TaoSetType() function or using the command line arguments -tao_type <name>. For those interested in extending these algorithms or using new ones, please see Adding a Solver for more information.

Unconstrained Minimization

Unconstrained minimization is used to minimize a function of many variables without any constraints on the variables, such as bounds. The methods available in TAO for solving these problems can be classified according to the amount of derivative information required:

1. Function evaluation only – Nelder-Mead method (tao_nm)

2. Function and gradient evaluations – limited-memory, variable-metric method (tao_lmvm) and nonlinear conjugate gradient method (tao_cg)

3. Function, gradient, and Hessian evaluations – Newton Krylov methods: Netwon line search (tao_nls), Newton trust-region (tao_ntr), and Newton trust-region line-search (tao_ntl)

The best method to use depends on the particular problem being solved and the accuracy required in the solution. If a Hessian evaluation routine is available, then the Newton line search and Newton trust-region methods will likely perform best. When a Hessian evaluation routine is not available, then the limited-memory, variable-metric method is likely to perform best. The Nelder-Mead method should be used only as a last resort when no gradient information is available.

Each solver has a set of options associated with it that can be set with command line arguments. These algorithms and the associated options are briefly discussed in this section.

Newton-Krylov Methods

TAO features three Newton-Krylov algorithms, separated by their globalization methods for unconstrained optimization: line search (NLS), trust region (NTR), and trust region with a line search (NTL). They are available via the TAO solvers TAONLS, TAONTR and TAONTL, respectively, or the -tao_type nls/ntr/ntl flag.

Newton Line Search Method (NLS)

The Newton line search method solves the symmetric system of equations

\[H_k d_k = -g_k \]

to obtain a step \(d_k\), where \(H_k\) is the Hessian of the objective function at \(x_k\) and \(g_k\) is the gradient of the objective function at \(x_k\). For problems where the Hessian matrix is indefinite, the perturbed system of equations

\[(H_k + \rho_k I) d_k = -g_k \]

is solved to obtain the direction, where \(\rho_k\) is a positive constant. If the direction computed is not a descent direction, the (scaled) steepest descent direction is used instead. Having obtained the direction, a Moré-Thuente line search is applied to obtain a step length, \(\tau_k\), that approximately solves the one-dimensional optimization problem

\[\min_\tau f(x_k + \tau d_k). \]

The Newton line search method can be selected by using the TAO solver tao_nls. The options available for this solver are listed in Table 18. For the best efficiency, function and gradient evaluations should be performed simultaneously when using this algorithm.

Table 18 Summary of nls options

Name	Value	Default	Description
-tao_nls_ksp_type	cg, nash, stcg, gltr, gmres, …	stcg	Type of Krylov subspace method to use when solving linear system
-tao_nls_pc_type	none, jacobi, icc, ilu, lmvm	lmvm	Type of preconditioner to use when solving linear system
-tao_nls_sval	real	\(0\)	Initial perturbation value
-tao_nls_imin	real	\(10^{-4}\)	Minimum initial perturbation value
-tao_nls_imax	real	\(100\)	Maximum initial perturbation value
-tao_nls_imfac	real	\(0.1\)	Factor applied to norm of gradient when initializing perturbation
-tao_nls_pmax	real	\(100\)	Maximum perturbation when increasing value
-tao_nls_pgfac	real	\(10\)	Growth factor applied to perturbation when increasing value
-tao_nls_pmgfac	real	\(0.1\)	Factor applied to norm of gradient when increasing perturbation
-tao_nls_pmin	real	\(10^{-12}\)	Minimum perturbation when decreasing value; smaller values set to zero
-tao_nls_psfac	real	\(0.4\)	Shrink factor applied to perturbation when decreasing value
-tao_nls_pmsfac	real	\(0.1\)	Factor applied to norm of gradient when decreasing perturbation
-tao_nls_init_type	constant, direction, interpolation	interpolation	Method used to initialize trust-region radius when using nash, stcg, or gltr
-tao_nls_mu1_i	real	0.35	\(\mu_1\) in interpolation init
-tao_nls_mu2_i	real	0.50	\(\mu_2\) in interpolation init
-tao_nls_gamma1_i	real	0.0625	\(\gamma_1\) in interpolation init
-tao_nls_gamma2_i	real	0.50	\(\gamma_2\) in interpolation init
-tao_nls_gamma3_i	real	2.00	\(\gamma_3\) in interpolation init
-tao_nls_gamma4_i	real	5.00	\(\gamma_4\) in interpolation init
-tao_nls_theta_i	real	0.25	\(\theta\) in interpolation init
-tao_nls_update_type	step, reduction, interpolation	step	Method used to update trust-region radius when using nash, stcg, or gltr
-tao_nls_nu1	real	0.25	\(\nu_1\) in step update
-tao_nls_nu2	real	0.50	\(\nu_2\) in step update
-tao_nls_nu3	real	1.00	\(\nu_3\) in step update
-tao_nls_nu4	real	1.25	\(\nu_4\) in step update
-tao_nls_omega1	real	0.25	\(\omega_1\) in step update
-tao_nls_omega2	real	0.50	\(\omega_2\) in step update
-tao_nls_omega3	real	1.00	\(\omega_3\) in step update
-tao_nls_omega4	real	2.00	\(\omega_4\) in step update
-tao_nls_omega5	real	4.00	\(\omega_5\) in step update
-tao_nls_eta1	real	\(10^{-4}\)	\(\eta_1\) in reduction update
-tao_nls_eta2	real	0.25	\(\eta_2\) in reduction update
-tao_nls_eta3	real	0.50	\(\eta_3\) in reduction update
-tao_nls_eta4	real	0.90	\(\eta_4\) in reduction update
-tao_nls_alpha1	real	0.25	\(\alpha_1\) in reduction update
-tao_nls_alpha2	real	0.50	\(\alpha_2\) in reduction update
-tao_nls_alpha3	real	1.00	\(\alpha_3\) in reduction update
-tao_nls_alpha4	real	2.00	\(\alpha_4\) in reduction update
-tao_nls_alpha5	real	4.00	\(\alpha_5\) in reduction update
-tao_nls_mu1	real	0.10	\(\mu_1\) in interpolation update
-tao_nls_mu2	real	0.50	\(\mu_2\) in interpolation update
-tao_nls_gamma1	real	0.25	\(\gamma_1\) in interpolation update
-tao_nls_gamma2	real	0.50	\(\gamma_2\) in interpolation update
-tao_nls_gamma3	real	2.00	\(\gamma_3\) in interpolation update
-tao_nls_gamma4	real	4.00	\(\gamma_4\) in interpolation update
-tao_nls_theta	real	0.05	\(\theta\) in interpolation update

The system of equations is approximately solved by applying the conjugate gradient method, Nash conjugate gradient method, Steihaug-Toint conjugate gradient method, generalized Lanczos method, or an alternative Krylov subspace method supplied by PETSc. The method used to solve the systems of equations is specified with the command line argument -tao_nls_ksp_type <cg,nash,stcg,gltr,gmres,...>; stcg is the default. See the PETSc manual for further information on changing the behavior of the linear system solvers.

A good preconditioner reduces the number of iterations required to solve the linear system of equations. For the conjugate gradient methods and generalized Lanczos method, this preconditioner must be symmetric and positive definite. The available options are to use no preconditioner, the absolute value of the diagonal of the Hessian matrix, a limited-memory BFGS approximation to the Hessian matrix, or one of the other preconditioners provided by the PETSc package. These preconditioners are specified by the command line arguments -tao_nls_pc_type <none,jacobi,icc,ilu,lmvm>, respectively. The default is the lmvm preconditioner, which uses a BFGS approximation of the inverse Hessian. See the PETSc manual for further information on changing the behavior of the preconditioners.

The perturbation \(\rho_k\) is added when the direction returned by the Krylov subspace method is not a descent direction, the Krylov method diverged due to an indefinite preconditioner or matrix, or a direction of negative curvature was found. In the last two cases, if the step returned is a descent direction, it is used during the line search. Otherwise, a steepest descent direction is used during the line search. The perturbation is decreased as long as the Krylov subspace method reports success and increased if further problems are encountered. There are three cases: initializing, increasing, and decreasing the perturbation. These cases are described below.

1. If \(\rho_k\) is zero and a problem was detected with either the direction or the Krylov subspace method, the perturbation is initialized to

   \[\rho_{k+1} = \text{median}\left\{\text{imin}, \text{imfac} * \|g(x_k)\|, \text{imax}\right\}, \]

   where \(g(x_k)\) is the gradient of the objective function and imin is set with the command line argument -tao_nls_imin <real> with a default value of \(10^{-4}\), imfac by -tao_nls_imfac with a default value of 0.1, and imax by -tao_nls_imax with a default value of 100. When using the gltr method to solve the system of equations, an estimate of the minimum eigenvalue \(\lambda_1\) of the Hessian matrix is available. This value is used to initialize the perturbation to \(\rho_{k+1} = \max\left\{\rho_{k+1}, -\lambda_1\right\}\) in this case.

2. If \(\rho_k\) is nonzero and a problem was detected with either the direction or Krylov subspace method, the perturbation is increased to

   \[\rho_{k+1} = \min\left\{\text{pmax}, \max\left\{\text{pgfac} * \rho_k, \text{pmgfac} * \|g(x_k)\|\right\}\right\}, \]

   where \(g(x_k)\) is the gradient of the objective function and pgfac is set with the command line argument -tao_nls_pgfac with a default value of 10, pmgfac by -tao_nls_pmgfac with a default value of 0.1, and pmax by -tao_nls_pmax with a default value of 100.

3. If \(\rho_k\) is nonzero and no problems were detected with either the direction or Krylov subspace method, the perturbation is decreased to

   \[\rho_{k+1} = \min\left\{\text{psfac} * \rho_k, \text{pmsfac} * \|g(x_k)\|\right\}, \]

   where \(g(x_k)\) is the gradient of the objective function, psfac is set with the command line argument -tao_nls_psfac with a default value of 0.4, and pmsfac is set by -tao_nls_pmsfac with a default value of 0.1. Moreover, if \(\rho_{k+1} < \text{pmin}\), then \(\rho_{k+1} = 0\), where pmin is set with the command line argument -tao_nls_pmin and has a default value of \(10^{-12}\).

Near a local minimizer to the unconstrained optimization problem, the Hessian matrix will be positive-semidefinite; the perturbation will shrink toward zero, and one would eventually observe a superlinear convergence rate.

When using nash, stcg, or gltr to solve the linear systems of equation, a trust-region radius needs to be initialized and updated. This trust-region radius simultaneously limits the size of the step computed and reduces the number of iterations of the conjugate gradient method. The method for initializing the trust-region radius is set with the command line argument -tao_nls_init_type <constant,direction,interpolation>; interpolation, which chooses an initial value based on the interpolation scheme found in [CGT00], is the default. This scheme performs a number of function and gradient evaluations to determine a radius such that the reduction predicted by the quadratic model along the gradient direction coincides with the actual reduction in the nonlinear function. The iterate obtaining the best objective function value is used as the starting point for the main line search algorithm. The constant method initializes the trust-region radius by using the value specified with the -tao_trust0 <real> command line argument, where the default value is 100. The direction technique solves the first quadratic optimization problem by using a standard conjugate gradient method and initializes the trust region to \(\|s_0\|\).

The method for updating the trust-region radius is set with the command line argument -tao_nls_update_type <step,reduction,interpolation>; step is the default. The step method updates the trust-region radius based on the value of \(\tau_k\). In particular,

\[\Delta_{k+1} = \left\{\begin{array}{ll} \omega_1 \text{min}(\Delta_k, \|d_k\|) & \text{if } \tau_k \in [0, \nu_1) \\ \omega_2 \text{min}(\Delta_k, \|d_k\|) & \text{if } \tau_k \in [\nu_1, \nu_2) \\ \omega_3 \Delta_k & \text{if } \tau_k \in [\nu_2, \nu_3) \\ \text{max}(\Delta_k, \omega_4 \|d_k\|) & \text{if } \tau_k \in [\nu_3, \nu_4) \\ \text{max}(\Delta_k, \omega_5 \|d_k\|) & \text{if } \tau_k \in [\nu_4, \infty), \end{array} \right.\]

where \(0 < \omega_1 < \omega_2 < \omega_3 = 1 < \omega_4 < \omega_5\) and \(0 < \nu_1 < \nu_2 < \nu_3 < \nu_4\) are constants. The reduction method computes the ratio of the actual reduction in the objective function to the reduction predicted by the quadratic model for the full step, \(\kappa_k = \frac{f(x_k) - f(x_k + d_k)}{q(x_k) - q(x_k + d_k)}\), where \(q_k\) is the quadratic model. The radius is then updated as

\[\Delta_{k+1} = \left\{\begin{array}{ll} \alpha_1 \text{min}(\Delta_k, \|d_k\|) & \text{if } \kappa_k \in (-\infty, \eta_1) \\ \alpha_2 \text{min}(\Delta_k, \|d_k\|) & \text{if } \kappa_k \in [\eta_1, \eta_2) \\ \alpha_3 \Delta_k & \text{if } \kappa_k \in [\eta_2, \eta_3) \\ \text{max}(\Delta_k, \alpha_4 \|d_k\|) & \text{if } \kappa_k \in [\eta_3, \eta_4) \\ \text{max}(\Delta_k, \alpha_5 \|d_k\|) & \text{if } \kappa_k \in [\eta_4, \infty), \end{array} \right.\]

where \(0 < \alpha_1 < \alpha_2 < \alpha_3 = 1 < \alpha_4 < \alpha_5\) and \(0 < \eta_1 < \eta_2 < \eta_3 < \eta_4\) are constants. The interpolation method uses the same interpolation mechanism as in the initialization to compute a new value for the trust-region radius.

This algorithm will be deprecated in the next version and replaced by the Bounded Newton Line Search (BNLS) algorithm that can solve both bound constrained and unconstrained problems.

Newton Trust-Region Method (NTR)

The Newton trust-region method solves the constrained quadratic programming problem

\[\begin{array}{ll} \min_d & \frac{1}{2}d^T H_k d + g_k^T d \\ \text{subject to} & \|d\| \leq \Delta_k \end{array}\]

to obtain a direction \(d_k\), where \(H_k\) is the Hessian of the objective function at \(x_k\), \(g_k\) is the gradient of the objective function at \(x_k\), and \(\Delta_k\) is the trust-region radius. If \(x_k + d_k\) sufficiently reduces the nonlinear objective function, then the step is accepted, and the trust-region radius is updated. However, if \(x_k + d_k\) does not sufficiently reduce the nonlinear objective function, then the step is rejected, the trust-region radius is reduced, and the quadratic program is re-solved by using the updated trust-region radius. The Newton trust-region method can be set by using the TAO solver tao_ntr. The options available for this solver are listed in Table 19. For the best efficiency, function and gradient evaluations should be performed separately when using this algorithm.

Table 19 Summary of ntr options

Name	Value	Default	Description
-tao_ntr_ksp_type	nash, stcg, gltr	stcg	Type of Krylov subspace method to use when solving linear system
-tao_ntr_pc_type	none, jacobi, icc, ilu, lmvm	lmvm	Type of preconditioner to use when solving linear system
-tao_ntr_init_type	constant, direction, interpolation	interpolation	Method used to initialize trust-region radius
-tao_ntr_mu1_i	real	0.35	\(\mu_1\) in interpolation init
-tao_ntr_mu2_i	real	0.50	\(\mu_2\) in interpolation init
-tao_ntr_gamma1_i	real	0.0625	\(\gamma_1\) in interpolation init
-tao_ntr_gamma2_i	real	0.50	\(\gamma_2\) in interpolation init
-tao_ntr_gamma3_i	real	2.00	\(\gamma_3\) in interpolation init
-tao_ntr_gamma4_i	real	5.00	\(\gamma_4\) in interpolation init
-tao_ntr_theta_i	real	0.25	\(\theta\) in interpolation init
-tao_ntr_update_type	reduction, interpolation	reduction	Method used to update trust-region radius
-tao_ntr_eta1	real	:	\(\eta_1\) in reduction update
-tao_ntr_eta2	real	0.25	\(\eta_2\) in reduction update
-tao_ntr_eta3	real	0.50	\(\eta_3\) in reduction update
-tao_ntr_eta4	real	0.90	\(\eta_4\) in reduction update
-tao_ntr_alpha1	real	0.25	\(\alpha_1\) in reduction update
-tao_ntr_alpha2	real	0.50	\(\alpha_2\) in reduction update
-tao_ntr_alpha3	real	1.00	\(\alpha_3\) in reduction update
-tao_ntr_alpha4	real	2.00	\(\alpha_4\) in reduction update
-tao_ntr_alpha5	real	4.00	\(\alpha_5\) in reduction update
-tao_ntr_mu1	real	0.10	\(\mu_1\) in interpolation update
-tao_ntr_mu2	real	0.50	\(\mu_2\) in interpolation update
-tao_ntr_gamma1	real	0.25	\(\gamma_1\) in interpolation update
-tao_ntr_gamma2	real	0.50	\(\gamma_2\) in interpolation update
-tao_ntr_gamma3	real	2.00	\(\gamma_3\) in interpolation update
-tao_ntr_gamma4	real	4.00	\(\gamma_4\) in interpolation update
-tao_ntr_theta	real	0.05	\(\theta\) in interpolation update

The quadratic optimization problem is approximately solved by applying the Nash or Steihaug-Toint conjugate gradient methods or the generalized Lanczos method to the symmetric system of equations \(H_k d = -g_k\). The method used to solve the system of equations is specified with the command line argument -tao_ntr_ksp_type <nash,stcg,gltr>; stcg is the default. See the PETSc manual for further information on changing the behavior of these linear system solvers.

A good preconditioner reduces the number of iterations required to compute the direction. For the Nash and Steihaug-Toint conjugate gradient methods and generalized Lanczos method, this preconditioner must be symmetric and positive definite. The available options are to use no preconditioner, the absolute value of the diagonal of the Hessian matrix, a limited-memory BFGS approximation to the Hessian matrix, or one of the other preconditioners provided by the PETSc package. These preconditioners are specified by the command line argument -tao_ntr_pc_type <none,jacobi,icc,ilu,lmvm>, respectively. The default is the lmvm preconditioner. See the PETSc manual for further information on changing the behavior of the preconditioners.

The method for computing an initial trust-region radius is set with the command line arguments -tao_ntr_init_type <constant,direction,interpolation>; interpolation, which chooses an initial value based on the interpolation scheme found in [CGT00], is the default. This scheme performs a number of function and gradient evaluations to determine a radius such that the reduction predicted by the quadratic model along the gradient direction coincides with the actual reduction in the nonlinear function. The iterate obtaining the best objective function value is used as the starting point for the main trust-region algorithm. The constant method initializes the trust-region radius by using the value specified with the -tao_trust0 <real> command line argument, where the default value is 100. The direction technique solves the first quadratic optimization problem by using a standard conjugate gradient method and initializes the trust region to \(\|s_0\|\).

The method for updating the trust-region radius is set with the command line arguments -tao_ntr_update_type <reduction,interpolation>; reduction is the default. The reduction method computes the ratio of the actual reduction in the objective function to the reduction predicted by the quadratic model for the full step, \(\kappa_k = \frac{f(x_k) - f(x_k + d_k)}{q(x_k) - q(x_k + d_k)}\), where \(q_k\) is the quadratic model. The radius is then updated as

\[\Delta_{k+1} = \left\{\begin{array}{ll} \alpha_1 \text{min}(\Delta_k, \|d_k\|) & \text{if } \kappa_k \in (-\infty, \eta_1) \\ \alpha_2 \text{min}(\Delta_k, \|d_k\|) & \text{if } \kappa_k \in [\eta_1, \eta_2) \\ \alpha_3 \Delta_k & \text{if } \kappa_k \in [\eta_2, \eta_3) \\ \text{max}(\Delta_k, \alpha_4 \|d_k\|) & \text{if } \kappa_k \in [\eta_3, \eta_4) \\ \text{max}(\Delta_k, \alpha_5 \|d_k\|) & \text{if } \kappa_k \in [\eta_4, \infty), \end{array} \right.\]

where \(0 < \alpha_1 < \alpha_2 < \alpha_3 = 1 < \alpha_4 < \alpha_5\) and \(0 < \eta_1 < \eta_2 < \eta_3 < \eta_4\) are constants. The interpolation method uses the same interpolation mechanism as in the initialization to compute a new value for the trust-region radius.

This algorithm will be deprecated in the next version and replaced by the Bounded Newton Trust Region (BNTR) algorithm that can solve both bound constrained and unconstrained problems.

Newton Trust Region with Line Search (NTL)

NTL safeguards the trust-region globalization such that a line search is used in the event that the step is initially rejected by the predicted versus actual decrease comparison. If the line search fails to find a viable step length for the Newton step, it falls back onto a scaled gradient or a gradient descent step. The trust radius is then modified based on the line search step length.

This algorithm will be deprecated in the next version and replaced by the Bounded Newton Trust Region with Line Search (BNTL) algorithm that can solve both bound constrained and unconstrained problems.

Limited-Memory Variable-Metric Method (LMVM)

The limited-memory, variable-metric method (LMVM) computes a positive definite approximation to the Hessian matrix from a limited number of previous iterates and gradient evaluations. A direction is then obtained by solving the system of equations

\[H_k d_k = -\nabla f(x_k), \]

where \(H_k\) is the Hessian approximation obtained by using the BFGS update formula. The inverse of \(H_k\) can readily be applied to obtain the direction \(d_k\). Having obtained the direction, a Moré-Thuente line search is applied to compute a step length, \(\tau_k\), that approximately solves the one-dimensional optimization problem

\[\min_\tau f(x_k + \tau d_k). \]

The current iterate and Hessian approximation are updated, and the process is repeated until the method converges. This algorithm is the default unconstrained minimization solver and can be selected by using the TAO solver tao_lmvm. For best efficiency, function and gradient evaluations should be performed simultaneously when using this algorithm.

The primary factors determining the behavior of this algorithm are the type of Hessian approximation used, the number of vectors stored for the approximation and the initialization/scaling of the approximation. These options can be configured using the -tao_lmvm_mat_lmvm prefix. For further detail, we refer the reader to the MATLMVM matrix type definitions in the PETSc Manual.

The LMVM algorithm also allows the user to define a custom initial Hessian matrix \(H_{0,k}\) through the interface function TaoLMVMSetH0(). This user-provided initialization overrides any other scalar or diagonal initialization inherent to the LMVM approximation. The provided \(H_{0,k}\) must be a PETSc Mat type object that represents a positive-definite matrix. The approximation prefers MatSolve() if the provided matrix has MATOP_SOLVE implemented. Otherwise, MatMult() is used in a KSP solve to perform the inversion of the user-provided initial Hessian.

In applications where TaoSolve() on the LMVM algorithm is repeatedly called to solve similar or related problems, -tao_lmvm_recycle flag can be used to prevent resetting the LMVM approximation between subsequent solutions. This recycling also avoids one extra function and gradient evaluation, instead re-using the values already computed at the end of the previous solution.

This algorithm will be deprecated in the next version and replaced by the Bounded Quasi-Newton Line Search (BQNLS) algorithm that can solve both bound constrained and unconstrained problems.

Nonlinear Conjugate Gradient Method (CG)

The nonlinear conjugate gradient method can be viewed as an extension of the conjugate gradient method for solving symmetric, positive-definite linear systems of equations. This algorithm requires only function and gradient evaluations as well as a line search. The TAO implementation uses a Moré-Thuente line search to obtain the step length. The nonlinear conjugate gradient method can be selected by using the TAO solver tao_cg. For the best efficiency, function and gradient evaluations should be performed simultaneously when using this algorithm.

Five variations are currently supported by the TAO implementation: the Fletcher-Reeves method, the Polak-Ribiére method, the Polak-Ribiére-Plus method [NW99], the Hestenes-Stiefel method, and the Dai-Yuan method. These conjugate gradient methods can be specified by using the command line argument -tao_cg_type <fr,pr,prp,hs,dy>, respectively. The default value is prp.

The conjugate gradient method incorporates automatic restarts when successive gradients are not sufficiently orthogonal. TAO measures the orthogonality by dividing the inner product of the gradient at the current point and the gradient at the previous point by the square of the Euclidean norm of the gradient at the current point. When the absolute value of this ratio is greater than \(\eta\), the algorithm restarts using the gradient direction. The parameter \(\eta\) can be set by using the command line argument -tao_cg_eta <real>; 0.1 is the default value.

This algorithm will be deprecated in the next version and replaced by the Bounded Nonlinear Conjugate Gradient (BNCG) algorithm that can solve both bound constrained and unconstrained problems.

Nelder-Mead Simplex Method (NM)

The Nelder-Mead algorithm [NM65] is a direct search method for finding a local minimum of a function \(f(x)\). This algorithm does not require any gradient or Hessian information of \(f\) and therefore has some expected advantages and disadvantages compared to the other TAO solvers. The obvious advantage is that it is easier to write an application when no derivatives need to be calculated. The downside is that this algorithm can be slow to converge or can even stagnate, and it performs poorly for large numbers of variables.

This solver keeps a set of \(N+1\) sorted vectors \({x_1,x_2,\ldots,x_{N+1}}\) and their corresponding objective function values \(f_1 \leq f_2 \leq \ldots \leq f_{N+1}\). At each iteration, \(x_{N+1}\) is removed from the set and replaced with

\[x(\mu) = (1+\mu) \frac{1}{N} \sum_{i=1}^N x_i - \mu x_{N+1}, \]

where \(\mu\) can be one of \({\mu_0,2\mu_0,\frac{1}{2}\mu_0,-\frac{1}{2}\mu_0}\) depending on the values of each possible \(f(x(\mu))\).

The algorithm terminates when the residual \(f_{N+1} - f_1\) becomes sufficiently small. Because of the way new vectors can be added to the sorted set, the minimum function value and/or the residual may not be impacted at each iteration.

Two options can be set specifically for the Nelder-Mead algorithm:

-tao_nm_lambda <value>

sets the initial set of vectors (\(x_0\) plus value in each coordinate direction); the default value is \(1\).

-tao_nm_mu <value>

sets the value of \(\mu_0\); the default is \(\mu_0=1\).

Bound-Constrained Optimization

Bound-constrained optimization algorithms solve optimization problems of the form

\[\begin{array}{ll} \displaystyle \min_{x} & f(x) \\ \text{subject to} & l \leq x \leq u. \end{array}\]

These solvers use the bounds on the variables as well as objective function, gradient, and possibly Hessian information.

For any unbounded variables, the bound value for the associated index can be set to PETSC_INFINITY for the upper bound and PETSC_NINFINITY for the lower bound. If all bounds are set to infinity, then the bounded algorithms are equivalent to their unconstrained counterparts.

Before introducing specific methods, we will first define two projection operations used by all bound constrained algorithms.

• Gradient projection:

  \[\mathfrak{P}(g) = \left\{\begin{array}{ll} 0 & \text{if} \; (x \leq l_i \land g_i > 0) \lor (x \geq u_i \land g_i < 0) \\ g_i & \text{otherwise} \end{array} \right.\]

• Bound projection:

  \[\mathfrak{B}(x) = \left\{\begin{array}{ll} l_i & \text{if} \; x_i < l_i \\ u_i & \text{if} \; x_i > u_i \\ x_i & \text{otherwise} \end{array} \right.\]

Bounded Newton-Krylov Methods

TAO features three bounded Newton-Krylov (BNK) class of algorithms, separated by their globalization methods: projected line search (BNLS), trust region (BNTR), and trust region with a projected line search fall-back (BNTL). They are available via the TAO solvers TAOBNLS, TAOBNTR and TAOBNTL, respectively, or the -tao_type bnls/bntr/bntl flag.

The BNK class of methods use an active-set approach to solve the symmetric system of equations,

\[H_k p_k = -g_k, \]

only for inactive variables in the interior of the bounds. The active-set estimation is based on Bertsekas [Ber82] with the following variable index categories:

\[\begin{array}{rlll} \displaystyle \text{lower bounded}: & \mathcal{L}(x) & = & \{ i \; : \; x_i \leq l_i + \epsilon \; \land \; g(x)_i > 0 \}, \\ \text{upper bounded}: & \mathcal{U}(x) & = & \{ i \; : \; x_i \geq u_i + \epsilon \; \land \; g(x)_i < 0 \}, \\ \text{fixed}: & \mathcal{F}(x) & = & \{ i \; : \; l_i = u_i \}, \\ \text{active-set}: & \mathcal{A}(x) & = & \{ \mathcal{L}(x) \; \bigcup \; \mathcal{U}(x) \; \bigcup \; \mathcal{F}(x) \}, \\ \text{inactive-set}: & \mathcal{I}(x) & = & \{ 1,2,\ldots,n \} \; \backslash \; \mathcal{A}(x). \end{array}\]

At each iteration, the bound tolerance is estimated as \(\epsilon_{k+1} = \text{min}(\epsilon_k, ||w_k||_2)\) with \(w_k = x_k - \mathfrak{B}(x_k - \beta D_k g_k)\), where the diagonal matrix \(D_k\) is an approximation of the Hessian inverse \(H_k^{-1}\). The initial bound tolerance \(\epsilon_0\) and the step length \(\beta\) have default values of \(0.001\) and can be adjusted using -tao_bnk_as_tol and -tao_bnk_as_step flags, respectively. The active-set estimation can be disabled using the option -tao_bnk_as_type none, in which case the algorithm simply uses the current iterate with no bound tolerances to determine which variables are actively bounded and which are free.

BNK algorithms invert the reduced Hessian using a Krylov iterative method. Trust-region conjugate gradient methods (KSPNASH, KSPSTCG, and KSPGLTR) are required for the BNTR and BNTL algorithms, and recommended for the BNLS algorithm. The preconditioner type can be changed using the -tao_bnk_pc_type none/ilu/icc/jacobi/lmvm. The lmvm option, which is also the default, preconditions the Krylov solution with a MATLMVM matrix. The remaining supported preconditioner types are default PETSc types. If Jacobi is selected, the diagonal values are safeguarded to be positive. icc and ilu options produce good results for problems with dense Hessians. The LMVM and Jacobi preconditioners are also used as the approximate inverse-Hessian in the active-set estimation. If neither are available, or if the Hessian matrix does not have MATOP_GET_DIAGONAL defined, then the active-set estimation falls back onto using an identity matrix in place of \(D_k\) (this is equivalent to estimating the active-set using a gradient descent step).

A special option is available to accelerate the convergence of the BNK algorithms by taking a finite number of BNCG iterations at each Newton iteration. By default, the number of BNCG iterations is set to zero and the algorithms do not take any BNCG steps. This can be changed using the option flag -tao_bnk_max_cg_its <i>. While this reduces the number of Newton iterations, in practice it simply trades off the Hessian evaluations in the BNK solver for more function and gradient evaluations in the BNCG solver. However, it may be useful for certain types of problems where the Hessian evaluation is disproportionately more expensive than the objective function or its gradient.

Bounded Newton Line Search (BNLS)

BNLS safeguards the Newton step by falling back onto a BFGS, scaled gradient, or gradient steps based on descent direction verifications. For problems with indefinite Hessian matrices, the step direction is calculated using a perturbed system of equations,

\[(H_k + \rho_k I)p_k = -g_k, \]

where \(\rho_k\) is a dynamically adjusted positive constant. The step is globalized using a projected Moré-Thuente line search. If a trust-region conjugate gradient method is used for the Hessian inversion, the trust radius is modified based on the line search step length.

Bounded Newton Trust Region (BNTR)

BNTR globalizes the Newton step using a trust region method based on the predicted versus actual reduction in the cost function. The trust radius is increased only if the accepted step is at the trust region boundary. The reduction check features a safeguard for numerical values below machine epsilon, scaled by the latest function value, where the full Newton step is accepted without modification.

Bounded Newton Trust Region with Line Search (BNTL)

BNTL safeguards the trust-region globalization such that a line search is used in the event that the step is initially rejected by the predicted versus actual decrease comparison. If the line search fails to find a viable step length for the Newton step, it falls back onto a scaled gradient or a gradient descent step. The trust radius is then modified based on the line search step length.

Bounded Quasi-Newton Line Search (BQNLS)

The BQNLS algorithm uses the BNLS infrastructure, but replaces the step calculation with a direct inverse application of the approximate Hessian based on quasi-Newton update formulas. No Krylov solver is used in the solution, and therefore the quasi-Newton method chosen must guarantee a positive-definite Hessian approximation. This algorithm is available via tao_type bqnls.

Bounded Quasi-Newton-Krylov

BQNK algorithms use the BNK infrastructure, but replace the exact Hessian with a quasi-Newton approximation. The matrix-free forward product operation based on quasi-Newton update formulas are used in conjunction with Krylov solvers to compute step directions. The quasi-Newton inverse application is used to precondition the Krylov solution, and typically helps converge to a step direction in \(\mathcal{O}(10)\) iterations. This approach is most useful with quasi-Newton update types such as Symmetric Rank-1 that cannot strictly guarantee positive-definiteness. The BNLS framework with Hessian shifting, or the BNTR framework with trust region safeguards, can successfully compensate for the Hessian approximation becoming indefinite.

Similar to the full Newton-Krylov counterpart, BQNK algorithms come in three forms separated by the globalization technique: line search (BQNKLS), trust region (BQNKTR) and trust region w/ line search fall-back (BQNKTL). These algorithms are available via tao_type <bqnkls, bqnktr, bqnktl>.

Bounded Nonlinear Conjugate Gradient (BNCG)

BNCG extends the unconstrained nonlinear conjugate gradient algorithm to bound constraints via gradient projections and a bounded Moré-Thuente line search.

Like its unconstrained counterpart, BNCG offers gradient descent and a variety of CG updates: Fletcher-Reeves, Polak-Ribiére, Polak-Ribiére-Plus, Hestenes-Stiefel, Dai-Yuan, Hager-Zhang, Dai-Kou, Kou-Dai, and the Self-Scaling Memoryless (SSML) BFGS, DFP, and Broyden methods. These methods can be specified by using the command line argument -tao_bncg_type <gd,fr,pr,prp,hs,dy,hz,dk,kd,ssml_bfgs,ssml_dfp,ssml_brdn>, respectively. The default value is ssml_bfgs. We have scalar preconditioning for these methods, and it is controlled by the flag tao_bncg_alpha. To disable rescaling, use \(\alpha = -1.0\), otherwise \(\alpha \in [0, 1]\). BNCG is available via the TAO solver TAOBNCG or the -tao_type bncg flag.

Some individual methods also contain their own parameters. The Hager-Zhang and Dou-Kai methods have a parameter that determines the minimum amount of contribution the previous search direction gives to the next search direction. The flags are -tao_bncg_hz_eta and -tao_bncg_dk_eta, and by default are set to \(0.4\) and \(0.5\) respectively. The Kou-Dai method has multiple parameters. -tao_bncg_zeta serves the same purpose as the previous two; set to \(0.1\) by default. There is also a parameter to scale the contribution of \(y_k \equiv \nabla f(x_k) - \nabla f(x_{k-1})\) in the search direction update. It is controlled by -tao_bncg_xi, and is equal to \(1.0\) by default. There are also times where we want to maximize the descent as measured by \(\nabla f(x_k)^T d_k\), and that may be done by using a negative value of \(\xi\); this achieves better performance when not using the diagonal preconditioner described next. This is enabled by default, and is controlled by -tao_bncg_neg_xi. Finally, the Broyden method has its convex combination parameter, set with -tao_bncg_theta. We have this as 1.0 by default, i.e. it is by default the BFGS method. One can also individually tweak the BFGS and DFP contributions using the multiplicative constants -tao_bncg_scale; both are set to \(1\) by default.

All methods can be scaled using the parameter -tao_bncg_alpha, which continuously varies in \([0, 1]\). The default value is set depending on the method from initial testing.

BNCG also offers a special type of method scaling. It employs Broyden diagonal scaling as an option for its CG methods, turned on with the flag -tao_bncg_diag_scaling. Formulations for both the forward (regular) and inverse Broyden methods are developed, controlled by the flag -tao_bncg_mat_lmvm_forward. It is set to True by default. Whether one uses the forward or inverse formulations depends on the method being used. For example, in our preliminary computations, the forward formulation works better for the SSML_BFGS method, but the inverse formulation works better for the Hestenes-Stiefel method. The convex combination parameter for the Broyden scaling is controlled by -tao_bncg_mat_lmvm_theta, and is 0 by default. We also employ rescaling of the Broyden diagonal, which aids the linesearch immensely. The rescaling parameter is controlled by -tao_bncg_mat_lmvm_alpha, and should be \(\in [0, 1]\). One can disable rescaling of the Broyden diagonal entirely by setting -tao_bncg_mat_lmvm_sigma_hist 0.

One can also supply their own preconditioner, serving as a Hessian initialization to the above diagonal scaling. The appropriate user function in the code is TaoBNCGSetH0(tao, H0) where H0 is the user-defined Mat object that serves as a preconditioner. For an example of similar usage, see tao/tutorials/ex3.c.

The active set estimation uses the Bertsekas-based method described in Bounded Newton-Krylov Methods, which can be deactivated using -tao_bncg_as_type none, in which case the algorithm will use the current iterate to determine the bounded variables with no tolerances and no look-ahead step. As in the BNK algorithm, the initial bound tolerance and estimator step length used in the Bertsekas method can be set via -tao_bncg_as_tol and -tao_bncg_as_step, respectively.

In addition to automatic scaled gradient descent restarts under certain local curvature conditions, we also employ restarts based on a check on descent direction such that \(\nabla f(x_k)^T d_k \in [-10^{11}, -10^{-9}]\). Furthermore, we allow for a variety of alternative restart strategies, all disabled by default. The -tao_bncg_unscaled_restart flag allows one to disable rescaling of the gradient for gradient descent steps. The -tao_bncg_spaced_restart flag tells the solver to restart every \(Mn\) iterations, where \(n\) is the problem dimension and \(M\) is a constant determined by -tao_bncg_min_restart_num and is 6 by default. We also have dynamic restart strategies based on checking if a function is locally quadratic; if so, go do a gradient descent step. The flag is -tao_bncg_dynamic_restart, disabled by default since the CG solver usually does better in those cases anyway. The minimum number of quadratic-like steps before a restart is set using -tao_bncg_min_quad and is 6 by default.

Generally Constrained Solvers

Constrained solvers solve optimization problems that incorporate either or both equality and inequality constraints, and may optionally include bounds on solution variables.

Alternating Direction Method of Multipliers (ADMM)

The TAOADMM algorithm is intended to blend the decomposability of dual ascent with the superior convergence properties of the method of multipliers. [BPC+11] The algorithm solves problems in the form

\[\begin{array}{ll} \displaystyle \min_{x} & f(x) + g(z) \\ \text{subject to} & Ax + Bz = c \end{array}\]

where \(x \in \mathbb R^n\), \(z \in \mathbb R^m\), \(A \in \mathbb R^{p \times n}\), \(B \in \mathbb R^{p \times m}\), and \(c \in \mathbb R^p\). Essentially, ADMM is a wrapper over two TAO solver, one for \(f(x)\), and one for \(g(z)\). With method of multipliers, one can form the augmented Lagrangian

\[L_{\rho}(x,z,y) = f(x) + g(z) + y^T(Ax+Bz-c) + (\rho/2)||Ax+Bz-c||_2^2 \]

Then, ADMM consists of the iterations

\[x^{k+1} := \text{argmin}L_{\rho}(x,z^k,y^k) \]

\[z^{k+1} := \text{argmin}L_{\rho}(x^{k+1},z,y^k) \]

\[y^{k+1} := y^k + \rho(Ax^{k+1}+Bz^{k+1}-c) \]

In certain formulation of ADMM, solution of \(z^{k+1}\) may have closed-form solution. Currently ADMM provides one default implementation for \(z^{k+1}\), which is soft-threshold. It can be used with either TaoADMMSetRegularizerType_ADMM() or -tao_admm_regularizer_type <regularizer_soft_thresh>. User can also pass spectral penalty value, \(\rho\), with either TaoADMMSetSpectralPenalty() or -tao_admm_spectral_penalty. Currently, user can use

• TaoADMMSetMisfitObjectiveAndGradientRoutine()

• TaoADMMSetRegularizerObjectiveAndGradientRoutine()

• TaoADMMSetMisfitHessianRoutine()

• TaoADMMSetRegularizerHessianRoutine()

Any other combination of routines is currently not supported. Hessian matrices can either be constant or non-constant, of which fact can be set via TaoADMMSetMisfitHessianChangeStatus(), and TaoADMMSetRegularizerHessianChangeStatus(). Also, it may appear in certain cases where augmented Lagrangian’s Hessian may become nearly singular depending on the \(\rho\), which may change in the case of -tao_admm_dual_update <update_adaptive>, <update_adaptive_relaxed>. This issue can be prevented by TaoADMMSetMinimumSpectralPenalty().

Augmented Lagrangian Method of Multipliers (ALMM)

The TAOALMM method solves generally constrained problems of the form

\[\begin{array}{ll} \displaystyle \min_{x} & f(x) \\ \text{subject to} & g(x) = 0\\ & h(x) \geq 0 \\ & l \leq x \leq u \end{array}\]

where \(g(x)\) are equality constraints, \(h(x)\) are inequality constraints and \(l\) and \(u\) are lower and upper bounds on the optimization variables, respectively.

TAOALMM converts the above general constrained problem into a sequence of bound constrained problems at each outer iteration \(k = 1,2,\dots\)

\[\begin{array}{ll} \displaystyle \min_{x} & L(x, \lambda_k) \\ \text{subject to} & l \leq x \leq u \end{array}\]

where \(L(x, \lambda_k)\) is the augmented Lagrangian merit function and \(\lambda_k\) is the Lagrange multiplier estimates at outer iteration \(k\).

TAOALMM offers two versions of the augmented Lagrangian formulation: the canonical Hestenes-Powell augmented Lagrangian [Hes69] [Pow69] with inequality constrained converted to equality constraints via slack variables, and the slack-less Powell-Hestenes-Rockafellar formulation [Roc74] that utilizes a pointwise max() on the inequality constraints. For most applications, the canonical Hestenes-Powell formulation is likely to perform better. However, the PHR formulation may be desirable for problems featuring very large numbers of inequality constraints as it avoids inflating the dimension of the subproblem with slack variables.

The inner subproblem is solved using a nested bound-constrained first-order TAO solver. By default, TAOALM uses a quasi-Newton-Krylov trust-region method (TAOBQNKTR). Other first-order methods such as TAOBNCG and TAOBQNLS are also appropriate, but a trust-region globalization is strongly recommended for most applications.

Primal-Dual Interior-Point Method (PDIPM)

The TAOPDIPM method (-tao_type pdipm) implements a primal-dual interior point method for solving general nonlinear programming problems of the form

(6)\[\begin{array}{ll} \displaystyle \min_{x} & f(x) \\ \text{subject to} & g(x) = 0 \\ & h(x) \geq 0 \\ & x^- \leq x \leq x^+ \end{array}\]

Here, \(f(x)\) is the nonlinear objective function, \(g(x)\), \(h(x)\) are the equality and inequality constraints, and \(x^-\) and \(x^+\) are the lower and upper bounds on decision variables \(x\).

PDIPM converts the inequality constraints to equalities using slack variables \(z\) and a log-barrier term, which transforms (6) to

(7)\[\begin{aligned} \text{min}~&f(x) - \mu\sum_{i=1}^{nci}\ln z_i\\ \text{s.t.}& \\ &ce(x) = 0 \\ &ci(x) - z = 0 \\ \end{aligned}\]

Here, \(ce(x)\) is set of equality constraints that include \(g(x)\) and fixed decision variables, i.e., \(x^- = x = x^+\). Similarly, \(ci(x)\) are inequality constraints including \(h(x)\) and lower/upper/box-constraints on \(x\). \(\mu\) is a parameter that is driven to zero as the optimization progresses.

The Lagrangian for (7)) is

(8)\[L_{\mu}(x,\lambda_{ce},\lambda_{ci},z) = f(x) + \lambda_{ce}^Tce(x) - \lambda_{ci}^T(ci(x) - z) - \mu\sum_{i=1}^{nci}\ln z_i\]

where, \(\lambda_{ce}\) and \(\lambda_{ci}\) are the Lagrangian multipliers for the equality and inequality constraints, respectively.

The first order KKT conditions for optimality are as follows

(9)\[\nabla L_{\mu}(x,\lambda_{ce},\lambda_{ci},z) = \begin{bmatrix} \nabla f(x) + \nabla ce(x)^T\lambda_{ce} - \nabla ci(x)^T \lambda_{ci} \\ ce(x) \\ ci(x) - z \\ Z\Lambda_{ci}e - \mu e \end{bmatrix} = 0\]

(9) is solved iteratively using Newton’s method using PETSc’s SNES object. After each Newton iteration, a line-search is performed to update \(x\) and enforce \(z,\lambda_{ci} \geq 0\). The barrier parameter \(\mu\) is also updated after each Newton iteration. The Newton update is obtained by solving the second-order KKT system \(Hd = -\nabla L_{\mu}\). Here,\(H\) is the Hessian matrix of the KKT system. For interior-point methods such as PDIPM, the Hessian matrix tends to be ill-conditioned, thus necessitating the use of a direct solver. We recommend using LU preconditioner -pc_type lu and using direct linear solver packages such SuperLU_Dist or MUMPS.

PDE-Constrained Optimization

TAO solves PDE-constrained optimization problems of the form

\[\begin{array}{ll} \displaystyle \min_{u,v} & f(u,v) \\ \text{subject to} & g(u,v) = 0, \end{array}\]

where the state variable \(u\) is the solution to the discretized partial differential equation defined by \(g\) and parametrized by the design variable \(v\), and \(f\) is an objective function. The Lagrange multipliers on the constraint are denoted by \(y\). This method is set by using the linearly constrained augmented Lagrangian TAO solver tao_lcl.

We make two main assumptions when solving these problems: the objective function and PDE constraints have been discretized so that we can treat the optimization problem as finite dimensional and \(\nabla_u g(u,v)\) is invertible for all \(u\) and \(v\).

Linearly-Constrained Augmented Lagrangian Method (LCL)

Given the current iterate \((u_k, v_k, y_k)\), the linearly constrained augmented Lagrangian method approximately solves the optimization problem

\[\begin{array}{ll} \displaystyle \min_{u,v} & \tilde{f}_k(u, v) \\ \text{subject to} & A_k (u-u_k) + B_k (v-v_k) + g_k = 0, \end{array}\]

where \(A_k = \nabla_u g(u_k,v_k)\), \(B_k = \nabla_v g(u_k,v_k)\), and \(g_k = g(u_k, v_k)\) and

\[\tilde{f}_k(u,v) = f(u,v) - g(u,v)^T y^k + \frac{\rho_k}{2} \| g(u,v) \|^2 \]

is the augmented Lagrangian function. This optimization problem is solved in two stages. The first computes the Newton direction and finds a feasible point for the linear constraints. The second computes a reduced-space direction that maintains feasibility with respect to the linearized constraints and improves the augmented Lagrangian merit function.

Newton Step

The Newton direction is obtained by fixing the design variables at their current value and solving the linearized constraint for the state variables. In particular, we solve the system of equations

\[A_k du = -g_k \]

to obtain a direction \(du\). We need a direction that provides sufficient descent for the merit function

\[\frac{1}{2} \|g(u,v)\|^2. \]

That is, we require \(g_k^T A_k du < 0\).

If the Newton direction is a descent direction, then we choose a penalty parameter \(\rho_k\) so that \(du\) is also a sufficient descent direction for the augmented Lagrangian merit function. We then find \(\alpha\) to approximately minimize the augmented Lagrangian merit function along the Newton direction.

\[\displaystyle \min_{\alpha \geq 0} \; \tilde{f}_k(u_k + \alpha du, v_k). \]

We can enforce either the sufficient decrease condition or the Wolfe conditions during the search procedure. The new point,

\[\begin{array}{lcl} u_{k+\frac{1}{2}} & = & u_k + \alpha_k du \\ v_{k+\frac{1}{2}} & = & v_k, \end{array}\]

satisfies the linear constraint

\[A_k (u_{k+\frac{1}{2}} - u_k) + B_k (v_{k+\frac{1}{2}} - v_k) + \alpha_k g_k = 0. \]

If the Newton direction computed does not provide descent for the merit function, then we can use the steepest descent direction \(du = -A_k^T g_k\) during the search procedure. However, the implication that the intermediate point approximately satisfies the linear constraint is no longer true.

Modified Reduced-Space Step

We are now ready to compute a reduced-space step for the modified optimization problem:

\[\begin{array}{ll} \displaystyle \min_{u,v} & \tilde{f}_k(u, v) \\ \text{subject to} & A_k (u-u_k) + B_k (v-v_k) + \alpha_k g_k = 0. \end{array}\]

We begin with the change of variables

\[\begin{array}{ll} \displaystyle \min_{du,dv} & \tilde{f}_k(u_k+du, v_k+dv) \\ \text{subject to} & A_k du + B_k dv + \alpha_k g_k = 0 \end{array}\]

and make the substitution

\[du = -A_k^{-1}(B_k dv + \alpha_k g_k). \]

Hence, the unconstrained optimization problem we need to solve is

\[\begin{array}{ll} \displaystyle \min_{dv} & \tilde{f}_k(u_k-A_k^{-1}(B_k dv + \alpha_k g_k), v_k+dv), \\ \end{array}\]

which is equivalent to

\[\begin{array}{ll} \displaystyle \min_{dv} & \tilde{f}_k(u_{k+\frac{1}{2}} - A_k^{-1} B_k dv, v_{k+\frac{1}{2}}+dv). \\ \end{array}\]

We apply one step of a limited-memory quasi-Newton method to this problem. The direction is obtain by solving the quadratic problem

\[\begin{array}{ll} \displaystyle \min_{dv} & \frac{1}{2} dv^T \tilde{H}_k dv + \tilde{g}_{k+\frac{1}{2}}^T dv, \end{array}\]

where \(\tilde{H}_k\) is the limited-memory quasi-Newton approximation to the reduced Hessian matrix, a positive-definite matrix, and \(\tilde{g}_{k+\frac{1}{2}}\) is the reduced gradient.

\[\begin{array}{lcl} \tilde{g}_{k+\frac{1}{2}} & = & \nabla_v \tilde{f}_k(u_{k+\frac{1}{2}}, v_{k+\frac{1}{2}}) - \nabla_u \tilde{f}_k(u_{k+\frac{1}{2}}, v_{k+\frac{1}{2}}) A_k^{-1} B_k \\ & = & d_{k+\frac{1}{2}} + c_{k+\frac{1}{2}} A_k^{-1} B_k \end{array}\]

The reduced gradient is obtained from one linearized adjoint solve

\[y_{k+\frac{1}{2}} = A_k^{-T}c_{k+\frac{1}{2}} \]

and some linear algebra

\[\tilde{g}_{k+\frac{1}{2}} = d_{k+\frac{1}{2}} + y_{k+\frac{1}{2}}^T B_k. \]

Because the Hessian approximation is positive definite and we know its inverse, we obtain the direction

\[dv = -H_k^{-1} \tilde{g}_{k+\frac{1}{2}} \]

and recover the full-space direction from one linearized forward solve,

\[du = -A_k^{-1} B_k dv. \]

Having the full-space direction, which satisfies the linear constraint, we now approximately minimize the augmented Lagrangian merit function along the direction.

\[\begin{array}{lcl} \displaystyle \min_{\beta \geq 0} & \tilde{f_k}(u_{k+\frac{1}{2}} + \beta du, v_{k+\frac{1}{2}} + \beta dv) \end{array}\]

We enforce the Wolfe conditions during the search procedure. The new point is

\[\begin{array}{lcl} u_{k+1} & = & u_{k+\frac{1}{2}} + \beta_k du \\ v_{k+1} & = & v_{k+\frac{1}{2}} + \beta_k dv. \end{array}\]

The reduced gradient at the new point is computed from

\[\begin{array}{lcl} y_{k+1} & = & A_k^{-T}c_{k+1} \\ \tilde{g}_{k+1} & = & d_{k+1} - y_{k+1}^T B_k, \end{array}\]

where \(c_{k+1} = \nabla_u \tilde{f}_k (u_{k+1},v_{k+1})\) and \(d_{k+1} = \nabla_v \tilde{f}_k (u_{k+1},v_{k+1})\). The multipliers \(y_{k+1}\) become the multipliers used in the next iteration of the code. The quantities \(v_{k+\frac{1}{2}}\), \(v_{k+1}\), \(\tilde{g}_{k+\frac{1}{2}}\), and \(\tilde{g}_{k+1}\) are used to update \(H_k\) to obtain the limited-memory quasi-Newton approximation to the reduced Hessian matrix used in the next iteration of the code. The update is skipped if it cannot be performed.

Nonlinear Least-Squares

Given a function \(F: \mathbb R^n \to \mathbb R^m\), the nonlinear least-squares problem minimizes

(10)\[f(x)= \| F(x) \|_2^2 = \sum_{i=1}^m F_i(x)^2.\]

The nonlinear equations \(F\) should be specified with the function TaoSetResidual().

Bound-constrained Regularized Gauss-Newton (BRGN)

The TAOBRGN algorithms is a Gauss-Newton method is used to iteratively solve nonlinear least squares problem with the iterations

\[x_{k+1} = x_k - \alpha_k(J_k^T J_k)^{-1} J_k^T r(x_k) \]

where \(r(x)\) is the least-squares residual vector, \(J_k = \partial r(x_k)/\partial x\) is the Jacobian of the residual, and \(\alpha_k\) is the step length parameter. In other words, the Gauss-Newton method approximates the Hessian of the objective as \(H_k \approx (J_k^T J_k)\) and the gradient of the objective as \(g_k \approx -J_k r(x_k)\). The least-squares Jacobian, \(J\), should be provided to Tao using TaoSetJacobianResidual() routine.

The BRGN (-tao_type brgn) implementation adds a regularization term \(\beta(x)\) such that

\[\min_{x} \; \frac{1}{2}||R(x)||_2^2 + \lambda\beta(x), \]

where \(\lambda\) is the scalar weight of the regularizer. BRGN provides two default implementations for \(\beta(x)\):

• L2-norm - \(\beta(x) = \frac{1}{2}||x_k||_2^2\)

• L2-norm Proximal Point - \(\beta(x) = \frac{1}{2}||x_k - x_{k-1}||_2^2\)

• L1-norm with Dictionary - \(\beta(x) = ||Dx||_1 \approx \sum_{i} \sqrt{y_i^2 + \epsilon^2}-\epsilon\) where \(y = Dx\) and \(\epsilon\) is the smooth approximation parameter.

The regularizer weight can be controlled with either TaoBRGNSetRegularizerWeight() or -tao_brgn_regularizer_weight command line option, while the smooth approximation parameter can be set with either TaoBRGNSetL1SmoothEpsilon() or -tao_brgn_l1_smooth_epsilon. For the L1-norm term, the user can supply a dictionary matrix with TaoBRGNSetDictionaryMatrix(). If no dictionary is provided, the dictionary is assumed to be an identity matrix and the regularizer reduces to a sparse solution term.

The regularization selection can be made using the command line option -tao_brgn_regularization_type <l2pure, l2prox, l1dict, user> where the user option allows the user to define a custom \(\mathcal{C}2\)-continuous regularization term. This custom term can be defined by using the interface functions:

• TaoBRGNSetRegularizerObjectiveAndGradientRoutine() - Provide user-call back for evaluating the function value and gradient evaluation for the regularization term.

• TaoBRGNSetRegularizerHessianRoutine() - Provide user call-back for evaluating the Hessian of the regularization term.

POUNDERS

One algorithm for solving the least squares problem ((10)) when the Jacobian of the residual vector \(F\) is unavailable is the model-based POUNDERS (Practical Optimization Using No Derivatives for sums of Squares) algorithm (tao_pounders). POUNDERS employs a derivative-free trust-region framework as described in [CSV09] in order to converge to local minimizers. An example of this version of POUNDERS applied to a practical least-squares problem can be found in [KortelainenLesinskiMore+10].

Derivative-Free Trust-Region Algorithm

In each iteration \(k\), the algorithm maintains a model \(m_k(x)\), described below, of the nonlinear least squares function \(f\) centered about the current iterate \(x_k\).

If one assumes that the maximum number of function evaluations has not been reached and that \(\|\nabla m_k(x_k)\|_2>\)gtol, the next point \(x_+\) to be evaluated is obtained by solving the trust-region subproblem

(11)\[\min\left\{ m_k(x) : \|x-x_k\|_{p} \leq \Delta_k, \right \},\]

where \(\Delta_k\) is the current trust-region radius. By default we use a trust-region norm with \(p=\infty\) and solve ((11)) with the BLMVM method described in Bound-constrained Limited-Memory Variable-Metric Method (BLMVM). While the subproblem is a bound-constrained quadratic program, it may not be convex and the BQPIP and GPCG methods may not solve the subproblem. Therefore, a bounded Newton-Krylov Method should be used; the default is the BNTR algorithm. Note: BNTR uses its own internal trust region that may interfere with the infinity-norm trust region used in the model problem ((11)).

The residual vector is then evaluated to obtain \(F(x_+)\) and hence \(f(x_+)\). The ratio of actual decrease to predicted decrease,

\[\rho_k = \frac{f(x_k)-f(x_+)}{m_k(x_k)-m_k(x_+)}, \]

as well as an indicator, valid, on the model’s quality of approximation on the trust region is then used to update the iterate,

\[x_{k+1} = \left\{\begin{array}{ll} x_+ & \text{if } \rho_k \geq \eta_1 \\ x_+ & \text{if } 0<\rho_k <\eta_1 \text{ and \texttt{valid}=\texttt{true}} \\ x_k & \text{else}, \end{array} \right.\]

and trust-region radius,

\[\Delta_{k+1} = \left\{\begin{array}{ll} \text{min}(\gamma_1\Delta_k, \Delta_{\max}) & \text{if } \rho_k \geq \eta_1 \text{ and } \|x_+-x_k\|_p\geq \omega_1\Delta_k \\ \gamma_0\Delta_k & \text{if } \rho_k < \eta_1 \text{ and \texttt{valid}=\texttt{true}} \\ \Delta_k & \text{else,} \end{array} \right.\]

where \(0 < \eta_1 < 1\), \(0 < \gamma_0 < 1 < \gamma_1\), \(0<\omega_1<1\), and \(\Delta_{\max}\) are constants.

If \(\rho_k\leq 0\) and valid is false, the iterate and trust-region radius remain unchanged after the above updates, and the algorithm tests whether the direction \(x_+-x_k\) improves the model. If not, the algorithm performs an additional evaluation to obtain \(F(x_k+d_k)\), where \(d_k\) is a model-improving direction.

The iteration counter is then updated, and the next model \(m_{k}\) is obtained as described next.

Forming the Trust-Region Model

In each iteration, POUNDERS uses a subset of the available evaluated residual vectors \(\{ F(y_1), F(y_2), \cdots \}\) to form an interpolatory quadratic model of each residual component. The \(m\) quadratic models

(12)\[q_k^{(i)}(x) = F_i(x_k) + (x-x_k)^T g_k^{(i)} + \frac{1}{2} (x-x_k)^T H_k^{(i)} (x-x_k), \qquad i = 1, \ldots, m\]

thus satisfy the interpolation conditions

\[q_k^{(i)}(y_j) = F_i(y_j), \qquad i=1, \ldots, m; \, j=1,\ldots , l_k \]

on a common interpolation set \(\{y_1, \cdots , y_{l_k}\}\) of size \(l_k\in[n+1,\)npmax\(]\).

The gradients and Hessians of the models in (:eq:eq_models) are then used to construct the main model,

(13)\[ m_k(x) = f(x_k) + 2(x-x_k)^T \sum_{i=1}^{m} F_i(x_k) g_k^{(i)} + (x-x_k)^T \sum_{i=1}^{m} \left( g_k^{(i)} \left(g_k^{(i)}\right)^T + F_i(x_k) H_k^{(i)}\right) (x-x_k).\]

The process of forming these models also computes the indicator valid of the model’s local quality.

Parameters

POUNDERS supports the following parameters that can be set from the command line or PETSc options file:

-tao_pounders_delta <delta>

The initial trust-region radius (\(>0\), real). This is used to determine the size of the initial neighborhood within which the algorithm should look.

-tao_pounders_npmax <npmax>

The maximum number of interpolation points used (\(n+2\leq\) npmax \(\leq 0.5(n+1)(n+2)\)). This input is made available to advanced users. We recommend the default value (npmax\(=2n+1\)) be used by others.

-tao_pounders_gqt

Use the gqt algorithm to solve the subproblem ((11)) (uses \(p=2\)) instead of BQPIP.

-pounders_subsolver

If the default BQPIP algorithm is used to solve the subproblem ((11)), the parameters of the subproblem solver can be accessed using the command line options prefix -pounders_subsolver_. For example,

-pounders_subsolver_tao_gatol 1.0e-5

sets the gradient tolerance of the subproblem solver to \(10^{-5}\).

Additionally, the user provides an initial solution vector, a vector for storing the separable objective function, and a routine for evaluating the residual vector \(F\). These are described in detail in Objective Function and Gradient Routines and Nonlinear Least Squares. Here we remark that because gradient information is not available for scaling purposes, it can be useful to ensure that the problem is reasonably well scaled. A simple way to do so is to rescale the decision variables \(x\) so that their typical values are expected to lie within the unit hypercube \([0,1]^n\).

Convergence Notes

Because the gradient function is not provided to POUNDERS, the norm of the gradient of the objective function is not available. Therefore, for convergence criteria, this norm is approximated by the norm of the model gradient and used only when the model gradient is deemed to be a reasonable approximation of the gradient of the objective. In practice, the typical grounds for termination for expensive derivative-free problems is the maximum number of function evaluations allowed.

Complementarity

Mixed complementarity problems, or box-constrained variational inequalities, are related to nonlinear systems of equations. They are defined by a continuously differentiable function, \(F:\mathbb R^n \to \mathbb R^n\), and bounds, \(\ell \in \{\mathbb R\cup \{-\infty\}\}^n\) and \(u \in \{\mathbb R\cup \{\infty\}\}^n\), on the variables such that \(\ell \leq u\). Given this information, \(\mathbf{x}^* \in [\ell,u]\) is a solution to MCP(\(F\), \(\ell\), \(u\)) if for each \(i \in \{1, \ldots, n\}\) we have at least one of the following:

\[\begin{aligned} \begin{array}{ll} F_i(x^*) \geq 0 & \text{if } x^*_i = \ell_i \\ F_i(x^*) = 0 & \text{if } \ell_i < x^*_i < u_i \\ F_i(x^*) \leq 0 & \text{if } x^*_i = u_i. \end{array}\end{aligned}\]

Note that when \(\ell = \{-\infty\}^n\) and \(u = \{\infty\}^n\), we have a nonlinear system of equations, and \(\ell = \{0\}^n\) and \(u = \{\infty\}^n\) correspond to the nonlinear complementarity problem [Cot64].

Simple complementarity conditions arise from the first-order optimality conditions from optimization [Kar39] [KT51]. In the simple bound-constrained optimization case, these conditions correspond to MCP(\(\nabla f\), \(\ell\), \(u\)), where \(f: \mathbb R^n \to \mathbb R\) is the objective function. In a one-dimensional setting these conditions are intuitive. If the solution is at the lower bound, then the function must be increasing and \(\nabla f \geq 0\). If the solution is at the upper bound, then the function must be decreasing and \(\nabla f \leq 0\). If the solution is strictly between the bounds, we must be at a stationary point and \(\nabla f = 0\). Other complementarity problems arise in economics and engineering [FP97], game theory [Nas50], and finance [HP98].

Evaluation routines for \(F\) and its Jacobian must be supplied prior to solving the application. The bounds, \([\ell,u]\), on the variables must also be provided. If no starting point is supplied, a default starting point of all zeros is used.

Semismooth Methods

TAO has two implementations of semismooth algorithms [MFF+01] [DeLucaFK96] [FFK97] for solving mixed complementarity problems. Both are based on a reformulation of the mixed complementarity problem as a nonsmooth system of equations using the Fischer-Burmeister function [Fis92]. A nonsmooth Newton method is applied to the reformulated system to calculate a solution. The theoretical properties of such methods are detailed in the aforementioned references.

The Fischer-Burmeister function, \(\phi:\mathbb R^2 \to \mathbb R\), is defined as

\[\begin{aligned} \phi(a,b) := \sqrt{a^2 + b^2} - a - b.\end{aligned}\]

This function has the following key property,

\[\begin{aligned} \begin{array}{lcr} \phi(a,b) = 0 & \Leftrightarrow & a \geq 0,\; b \geq 0,\; ab = 0, \end{array}\end{aligned}\]

used when reformulating the mixed complementarity problem as the system of equations \(\Phi(x) = 0\), where \(\Phi:\mathbb R^n \to \mathbb R^n\). The reformulation is defined componentwise as

\[\begin{aligned} \Phi_i(x) := \left\{ \begin{array}{ll} \phi(x_i - l_i, F_i(x)) & \text{if } -\infty < l_i < u_i = \infty, \\ -\phi(u_i-x_i, -F_i(x)) & \text{if } -\infty = l_i < u_i < \infty, \\ \phi(x_i - l_i, \phi(u_i - x_i, - F_i(x))) & \text{if } -\infty < l_i < u_i < \infty, \\ -F_i(x) & \text{if } -\infty = l_i < u_i = \infty, \\ l_i - x_i & \text{if } -\infty < l_i = u_i < \infty. \end{array} \right.\end{aligned}\]

We note that \(\Phi\) is not differentiable everywhere but satisfies a semismoothness property [Mif77] [Qi93] [QS93]. Furthermore, the natural merit function, \(\Psi(x) := \frac{1}{2} \| \Phi(x) \|_2^2\), is continuously differentiable.

The two semismooth TAO solvers both solve the system \(\Phi(x) = 0\) by applying a nonsmooth Newton method with a line search. We calculate a direction, \(d^k\), by solving the system \(H^kd^k = -\Phi(x^k)\), where \(H^k\) is an element of the \(B\)-subdifferential [QS93] of \(\Phi\) at \(x^k\). If the direction calculated does not satisfy a suitable descent condition, then we use the negative gradient of the merit function, \(-\nabla \Psi(x^k)\), as the search direction. A standard Armijo search [Arm66] is used to find the new iteration. Nonmonotone searches [GLL86] are also available by setting appropriate runtime options. See Line Searches for further details.

The first semismooth algorithm available in TAO is not guaranteed to remain feasible with respect to the bounds, \([\ell, u]\), and is termed an infeasible semismooth method. This method can be specified by using the tao_ssils solver. In this case, the descent test used is that

\[\begin{aligned} \nabla \Psi(x^k)^Td^k \leq -\delta\| d^k \|^\rho.\end{aligned}\]

Both \(\delta > 0\) and \(\rho > 2\) can be modified by using the runtime options -tao_ssils_delta <delta> and -tao_ssils_rho <rho>, respectively. By default, \(\delta = 10^{-10}\) and \(\rho = 2.1\).

An alternative is to remain feasible with respect to the bounds by using a projected Armijo line search. This method can be specified by using the tao_ssfls solver. The descent test used is the same as above where the direction in this case corresponds to the first part of the piecewise linear arc searched by the projected line search. Both \(\delta > 0\) and \(\rho > 2\) can be modified by using the runtime options -tao_ssfls_delta <delta> and -tao_ssfls_rho <rho> respectively. By default, \(\delta = 10^{-10}\) and \(\rho = 2.1\).

The recommended algorithm is the infeasible semismooth method, tao_ssils, because of its strong global and local convergence properties. However, if it is known that \(F\) is not defined outside of the box, \([\ell,u]\), perhaps because of the presence of \(\log\) functions, the feasibility-enforcing version of the algorithm, tao_ssfls, is a reasonable alternative.

Active-Set Methods

TAO also contained two active-set semismooth methods for solving complementarity problems. These methods solve a reduced system constructed by block elimination of active constraints. The subdifferential in these cases enables this block elimination.

The first active-set semismooth algorithm available in TAO is not guaranteed to remain feasible with respect to the bounds, \([\ell, u]\), and is termed an infeasible active-set semismooth method. This method can be specified by using the tao_asils solver.

An alternative is to remain feasible with respect to the bounds by using a projected Armijo line search. This method can be specified by using the tao_asfls solver.

Quadratic Solvers

Quadratic solvers solve optimization problems of the form

\[\begin{array}{ll} \displaystyle \min_{x} & \frac{1}{2}x^T Q x + c^T x \\ \text{subject to} & l \geq x \geq u \end{array}\]

where the gradient and the Hessian of the objective are both constant.

Gradient Projection Conjugate Gradient Method (GPCG)

The GPCG [MoreT91] algorithm is much like the TRON algorithm, discussed in Section Trust-Region Newton Method (TRON), except that it assumes that the objective function is quadratic and convex. Therefore, it evaluates the function, gradient, and Hessian only once. Since the objective function is quadratic, the algorithm does not use a trust region. All the options that apply to TRON except for trust-region options also apply to GPCG. It can be set by using the TAO solver tao_gpcg or via the optio flag -tao_type gpcg.

Interior-Point Newton’s Method (BQPIP)

The BQPIP algorithm is an interior-point method for bound constrained quadratic optimization. It can be set by using the TAO solver of tao_bqpip or via the option flag -tao_type bgpip. Since it assumes the objective function is quadratic, it evaluates the function, gradient, and Hessian only once. This method also requires the solution of systems of linear equations, whose solver can be accessed and modified with the command TaoGetKSP().

Legacy and Contributed Solvers

Bundle Method for Regularized Risk Minimization (BMRM)

BMRM is a numerical approach to optimizing an unconstrained objective in the form of \(f(x) + 0.5 * \lambda \| x \|^2\). Here \(f\) is a convex function that is finite on the whole space. \(\lambda\) is a positive weight parameter, and \(\| x \|\) is the Euclidean norm of \(x\). The algorithm only requires a routine which, given an \(x\), returns the value of \(f(x)\) and the gradient of \(f\) at \(x\).

Orthant-Wise Limited-memory Quasi-Newton (OWLQN)

OWLQN [AG07] is a numerical approach to optimizing an unconstrained objective in the form of \(f(x) + \lambda \|x\|_1\). Here f is a convex and differentiable function, \(\lambda\) is a positive weight parameter, and \(\| x \|_1\) is the \(\ell_1\) norm of \(x\): \(\sum_i |x_i|\). The algorithm only requires evaluating the value of \(f\) and its gradient.

Trust-Region Newton Method (TRON)

The TRON [LMore99] algorithm is an active-set method that uses a combination of gradient projections and a preconditioned conjugate gradient method to minimize an objective function. Each iteration of the TRON algorithm requires function, gradient, and Hessian evaluations. In each iteration, the algorithm first applies several conjugate gradient iterations. After these iterates, the TRON solver momentarily ignores the variables that equal one of its bounds and applies a preconditioned conjugate gradient method to a quadratic model of the remaining set of free variables.

The TRON algorithm solves a reduced linear system defined by the rows and columns corresponding to the variables that lie between the upper and lower bounds. The TRON algorithm applies a trust region to the conjugate gradients to ensure convergence. The initial trust-region radius can be set by using the command TaoSetInitialTrustRegionRadius(), and the current trust region size can be found by using the command TaoGetCurrentTrustRegionRadius(). The initial trust region can significantly alter the rate of convergence for the algorithm and should be tuned and adjusted for optimal performance.

This algorithm will be deprecated in the next version in favor of the Bounded Newton Trust Region (BNTR) algorithm.

Bound-constrained Limited-Memory Variable-Metric Method (BLMVM)

BLMVM is a limited-memory, variable-metric method and is the bound-constrained variant of the LMVM method for unconstrained optimization. It uses projected gradients to approximate the Hessian, eliminating the need for Hessian evaluations. The method can be set by using the TAO solver tao_blmvm. For more details, please see the LMVM section in the unconstrained algorithms as well as the LMVM matrix documentation in the PETSc manual.

This algorithm will be deprecated in the next version in favor of the Bounded Quasi-Newton Line Search (BQNLS) algorithm.

Advanced Options

This section discusses options and routines that apply to most TAO solvers and problem classes. In particular, we focus on linear solvers, convergence tests, and line searches.

Linear Solvers

One of the most computationally intensive phases of many optimization algorithms involves the solution of linear systems of equations. The performance of the linear solver may be critical to an efficient computation of the solution. Since linear equation solvers often have a wide variety of options associated with them, TAO allows the user to access the linear solver with the

TaoGetKSP(Tao, KSP *);

command. With access to the KSP object, users can customize it for their application to achieve improved performance. Additional details on the KSP options in PETSc can be found in the User-Guide.

Monitors

By default the TAO solvers run silently without displaying information about the iterations. The user can initiate monitoring with the command

TaoSetMonitor(Tao, PetscErrorCode (*mon)(Tao,void*), void*);

The routine mon indicates a user-defined monitoring routine, and void* denotes an optional user-defined context for private data for the monitor routine.

The routine set by TaoSetMonitor() is called once during each iteration of the optimization solver. Hence, the user can employ this routine for any application-specific computations that should be done after the solution update.

Convergence Tests

Convergence of a solver can be defined in many ways. The methods TAO uses by default are mentioned in Convergence. These methods include absolute and relative convergence tolerances as well as a maximum number of iterations of function evaluations. If these choices are not sufficient, the user can specify a customized test

Users can set their own customized convergence tests of the form

PetscErrorCode  conv(Tao, void*);

The second argument is a pointer to a structure defined by the user. Within this routine, the solver can be queried for the solution vector, gradient vector, or other statistic at the current iteration through routines such as TaoGetSolutionStatus() and TaoGetTolerances().

To use this convergence test within a TAO solver, one uses the command

TaoSetConvergenceTest(Tao, PetscErrorCode (*conv)(Tao,void*), void*);

The second argument of this command is the convergence routine, and the final argument of the convergence test routine denotes an optional user-defined context for private data. The convergence routine receives the TAO solver and this private data structure. The termination flag can be set by using the routine

TaoSetConvergedReason(Tao, TaoConvergedReason);

Line Searches

By using the command line option -tao_ls_type. Available line searches include Moré-Thuente [MoreT92], Armijo, gpcg, and unit.

The line search routines involve several parameters, which are set to defaults that are reasonable for many applications. The user can override the defaults by using the following options

• -tao_ls_max_funcs <max>

• -tao_ls_stepmin <min>

• -tao_ls_stepmax <max>

• -tao_ls_ftol <ftol>

• -tao_ls_gtol <gtol>

• -tao_ls_rtol <rtol>

One should run a TAO program with the option -help for details. Users may write their own customized line search codes by modeling them after one of the defaults provided.

Recycling History

Some TAO algorithms can re-use information accumulated in the previous TaoSolve() call to hot-start the new solution. This can be enabled using the -tao_recycle_history flag, or in code via the TaoSetRecycleHistory() interface.

For the nonlinear conjugate gradient solver (TAOBNCG), this option re-uses the latest search direction from the previous TaoSolve() call to compute the initial search direction of a new TaoSolve(). By default, the feature is disabled and the algorithm sets the initial direction as the negative gradient.

For the quasi-Newton family of methods (TAOBQNLS, TAOBQNKLS, TAOBQNKTR, TAOBQNKTL), this option re-uses the accumulated quasi-Newton Hessian approximation from the previous TaoSolve() call. By default, the feature is disabled and the algorithm will reset the quasi-Newton approximation to the identity matrix at the beginning of every new TaoSolve().

The option flag has no effect on other TAO solvers.

Adding a Solver

One of the strengths of both TAO and PETSc is the ability to allow users to extend the built-in solvers with new user-defined algorithms. It is certainly possible to develop new optimization algorithms outside of TAO framework, but Using TAO to implement a solver has many advantages,

1. TAO includes other optimization solvers with an identical interface, so application problems may conveniently switch solvers to compare their effectiveness.

2. TAO provides support for function evaluations and derivative information. It allows for the direct evaluation of this information by the application developer, contains limited support for finite difference approximations, and allows the uses of matrix-free methods. The solvers can obtain this function and derivative information through a simple interface while the details of its computation are handled within the toolkit.

3. TAO provides line searches, convergence tests, monitoring routines, and other tools that are helpful in an optimization algorithm. The availability of these tools means that the developers of the optimization solver do not have to write these utilities.

4. PETSc offers vectors, matrices, index sets, and linear solvers that can be used by the solver. These objects are standard mathematical constructions that have many different implementations. The objects may be distributed over multiple processors, restricted to a single processor, have a dense representation, use a sparse data structure, or vary in many other ways. TAO solvers do not need to know how these objects are represented or how the operations defined on them have been implemented. Instead, the solvers apply these operations through an abstract interface that leaves the details to PETSc and external libraries. This abstraction allows solvers to work seamlessly with a variety of data structures while allowing application developers to select data structures tailored for their purposes.

5. PETSc provides the user a convenient method for setting options at runtime, performance profiling, and debugging.

Header File

TAO solver implementation files must include the TAO implementation file taoimpl.h:

#include "petsc/private/taoimpl.h"

This file contains data elements that are generally kept hidden from application programmers, but may be necessary for solver implementations to access.

TAO Interface with Solvers

TAO solvers must be written in C or C++ and include several routines with a particular calling sequence. Two of these routines are mandatory: one that initializes the TAO structure with the appropriate information and one that applies the algorithm to a problem instance. Additional routines may be written to set options within the solver, view the solver, setup appropriate data structures, and destroy these data structures. In order to implement the conjugate gradient algorithm, for example, the following structure is useful.

typedef struct{

  PetscReal beta;
  PetscReal eta;
  PetscInt  ngradtseps;
  PetscInt  nresetsteps;
  Vec X_old;
  Vec G_old;

} TAO_CG;

This structure contains two parameters, two counters, and two work vectors. Vectors for the solution and gradient are not needed here because the TAO structure has pointers to them.

Solver Routine

All TAO solvers have a routine that accepts a TAO structure and computes a solution. TAO will call this routine when the application program uses the routine TaoSolve() and will pass to the solver information about the objective function and constraints, pointers to the variable vector and gradient vector, and support for line searches, linear solvers, and convergence monitoring. As an example, consider the following code that solves an unconstrained minimization problem using the conjugate gradient method.

PetscErrorCode TaoSolve_CG(Tao tao)
{
  TAO_CG  *cg = (TAO_CG *) tao->data;
  Vec x = tao->solution;
  Vec g = tao->gradient;
  Vec s = tao->stepdirection;
  PetscInt     iter=0;
  PetscReal  gnormPrev,gdx,f,gnorm,steplength=0;
  TaoLineSearchConvergedReason lsflag=TAO_LINESEARCH_CONTINUE_ITERATING;
  TaoConvergedReason reason=TAO_CONTINUE_ITERATING;

  PetscFunctionBegin;

  PetscCall(TaoComputeObjectiveAndGradient(tao,x,&f,g));
  PetscCall(VecNorm(g,NORM_2,&gnorm));

  PetscCall(VecSet(s,0));

  cg->beta=0;
  gnormPrev = gnorm;

  /* Enter loop */
  while (1){

    /* Test for convergence */
    PetscCall(TaoMonitor(tao,iter,f,gnorm,0.0,step,&reason));
    if (reason!=TAO_CONTINUE_ITERATING) break;

    cg->beta=(gnorm*gnorm)/(gnormPrev*gnormPrev);
    PetscCall(VecScale(s,cg->beta));
    PetscCall(VecAXPY(s,-1.0,g));

    PetscCall(VecDot(s,g,&gdx));
    if (gdx>=0){     /* If not a descent direction, use gradient */
      PetscCall(VecCopy(g,s));
      PetscCall(VecScale(s,-1.0));
      gdx=-gnorm*gnorm;
    }

    /* Line Search */
    gnormPrev = gnorm;  step=1.0;
    PetscCall(TaoLineSearchSetInitialStepLength(tao->linesearch,1.0));
    PetscCall(TaoLineSearchApply(tao->linesearch,x,&f,g,s,&steplength,&lsflag));
    PetscCall(TaoAddLineSearchCounts(tao));
    PetscCall(VecNorm(g,NORM_2,&gnorm));
    iter++;
  }

  PetscFunctionReturn(PETSC_SUCCESS);
}

The first line of this routine casts the second argument to a pointer to a TAO_CG data structure. This structure contains pointers to three vectors and a scalar that will be needed in the algorithm.

After declaring an initializing several variables, the solver lets TAO evaluate the function and gradient at the current point in the using the routine TaoComputeObjectiveAndGradient(). Other routines may be used to evaluate the Hessian matrix or evaluate constraints. TAO may obtain this information using direct evaluation or other means, but these details do not affect our implementation of the algorithm.

The norm of the gradient is a standard measure used by unconstrained minimization solvers to define convergence. This quantity is always nonnegative and equals zero at the solution. The solver will pass this quantity, the current function value, the current iteration number, and a measure of infeasibility to TAO with the routine

PetscErrorCode TaoMonitor(Tao tao, PetscInt iter, PetscReal f,
               PetscReal res, PetscReal cnorm, PetscReal steplength,
               TaoConvergedReason *reason);

Most optimization algorithms are iterative, and solvers should include this command somewhere in each iteration. This routine records this information, and applies any monitoring routines and convergence tests set by default or the user. In this routine, the second argument is the current iteration number, and the third argument is the current function value. The fourth argument is a nonnegative error measure associated with the distance between the current solution and the optimal solution. Examples of this measure are the norm of the gradient or the square root of a duality gap. The fifth argument is a nonnegative error that usually represents a measure of the infeasibility such as the norm of the constraints or violation of bounds. This number should be zero for unconstrained solvers. The sixth argument is a nonnegative steplength, or the multiple of the step direction added to the previous iterate. The results of the convergence test are returned in the last argument. If the termination reason is TAO_CONTINUE_ITERATING, the algorithm should continue.

After this monitoring routine, the solver computes a step direction using the conjugate gradient algorithm and computations using Vec objects. These methods include adding vectors together and computing an inner product. A full list of these methods can be found in the manual pages.

Nonlinear conjugate gradient algorithms also require a line search. TAO provides several line searches and support for using them. The routine

TaoLineSearchApply(TaoLineSearch ls, Vec x, PetscReal *f, Vec g,
                       TaoVec *s, PetscReal *steplength,
                       TaoLineSearchConvergedReason *lsflag)

passes the current solution, gradient, and objective value to the line search and returns a new solution, gradient, and objective value. More details on line searches can be found in Line Searches. The details of the line search applied are specified elsewhere, when the line search is created.

TAO also includes support for linear solvers using PETSc KSP objects. Although this algorithm does not require one, linear solvers are an important part of many algorithms. Details on the use of these solvers can be found in the PETSc users manual.

Creation Routine

The TAO solver is initialized for a particular algorithm in a separate routine. This routine sets default convergence tolerances, creates a line search or linear solver if needed, and creates structures needed by this solver. For example, the routine that creates the nonlinear conjugate gradient algorithm shown above can be implemented as follows.

PETSC_EXTERN PetscErrorCode TaoCreate_CG(Tao tao)
{
  TAO_CG *cg = (TAO_CG*)tao->data;
  const char *morethuente_type = TAOLINESEARCH_MT;

  PetscFunctionBegin;

  PetscCall(PetscNew(&cg));
  tao->data = (void*)cg;
  cg->eta = 0.1;
  cg->delta_min = 1e-7;
  cg->delta_max = 100;
  cg->cg_type = CG_PolakRibierePlus;

  tao->max_it = 2000;
  tao->max_funcs = 4000;

  tao->ops->setup = TaoSetUp_CG;
  tao->ops->solve = TaoSolve_CG;
  tao->ops->view = TaoView_CG;
  tao->ops->setfromoptions = TaoSetFromOptions_CG;
  tao->ops->destroy = TaoDestroy_CG;

  PetscCall(TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch));
  PetscCall(TaoLineSearchSetType(tao->linesearch, morethuente_type));
  PetscCall(TaoLineSearchUseTaoRoutines(tao->linesearch, tao));

  PetscFunctionReturn(PETSC_SUCCESS);
}
EXTERN_C_END

This routine declares some variables and then allocates memory for the TAO_CG data structure. Notice that the Tao object now has a pointer to this data structure (tao->data) so it can be accessed by the other functions written for this solver implementation.

This routine also sets some default parameters particular to the conjugate gradient algorithm, sets default convergence tolerances, and creates a particular line search. These defaults could be specified in the routine that solves the problem, but specifying them here gives the user the opportunity to modify these parameters either by using direct calls setting parameters or by using options.

Finally, this solver passes to TAO the names of all the other routines used by the solver.

Note that the lines EXTERN_C_BEGIN and EXTERN_C_END surround this routine. These macros are required to preserve the name of this function without any name-mangling from the C++ compiler (if used).

Destroy Routine

Another routine needed by most solvers destroys the data structures created by earlier routines. For the nonlinear conjugate gradient method discussed earlier, the following routine destroys the two work vectors and the TAO_CG structure.

PetscErrorCode TaoDestroy_CG(TAO_SOLVER tao)
{
  TAO_CG *cg = (TAO_CG *) tao->data;

  PetscFunctionBegin;

  PetscCall(VecDestroy(&cg->X_old));
  PetscCall(VecDestroy(&cg->G_old));

  PetscFree(tao->data);
  tao->data = NULL;

  PetscFunctionReturn(PETSC_SUCCESS);
}

This routine is called from within the TaoDestroy() routine. Only algorithm-specific data objects are destroyed in this routine; any objects indexed by TAO (tao->linesearch, tao->ksp, tao->gradient, etc.) will be destroyed by TAO immediately after the algorithm-specific destroy routine completes.

SetUp Routine

If the SetUp routine has been set by the initialization routine, TAO will call it during the execution of TaoSolve(). While this routine is optional, it is often provided to allocate the gradient vector, work vectors, and other data structures required by the solver. It should have the following form.

PetscErrorCode TaoSetUp_CG(Tao tao)
{
  TAO_CG *cg = (TAO_CG*)tao->data;
  PetscFunctionBegin;

  PetscCall(VecDuplicate(tao->solution,&tao->gradient));
  PetscCall(VecDuplicate(tao->solution,&tao->stepdirection));
  PetscCall(VecDuplicate(tao->solution,&cg->X_old));
  PetscCall(VecDuplicate(tao->solution,&cg->G_old));

  PetscFunctionReturn(PETSC_SUCCESS);
}

SetFromOptions Routine

The SetFromOptions routine should be used to check for any algorithm-specific options set by the user and will be called when the application makes a call to TaoSetFromOptions(). It should have the following form.

PetscErrorCode TaoSetFromOptions_CG(Tao tao, void *solver);
{
  TAO_CG *cg = (TAO_CG*)solver;
  PetscFunctionBegin;
  PetscCall(PetscOptionsReal("-tao_cg_eta","restart tolerance","",cg->eta,&cg->eta,0));
  PetscCall(PetscOptionsReal("-tao_cg_delta_min","minimum delta value","",cg->delta_min,&cg->delta_min,0));
  PetscCall(PetscOptionsReal("-tao_cg_delta_max","maximum delta value","",cg->delta_max,&cg->delta_max,0));
  PetscFunctionReturn(PETSC_SUCCESS);
}

View Routine

The View routine should be used to output any algorithm-specific information or statistics at the end of a solve. This routine will be called when the application makes a call to TaoView() or when the command line option -tao_view is used. It should have the following form.

PetscErrorCode TaoView_CG(Tao tao, PetscViewer viewer)
{
  TAO_CG *cg = (TAO_CG*)tao->data;

  PetscFunctionBegin;
  PetscCall(PetscViewerASCIIPushTab(viewer));
  PetscCall(PetscViewerASCIIPrintf(viewer,"Grad. steps: %d\n",cg->ngradsteps));
  PetscCall(PetscViewerASCIIPrintf(viewer,"Reset steps: %d\n",cg->nresetsteps));
  PetscCall(PetscViewerASCIIPopTab(viewer));
  PetscFunctionReturn(PETSC_SUCCESS);
}

Registering the Solver

Once a new solver is implemented, TAO needs to know the name of the solver and what function to use to create the solver. To this end, one can use the routine

TaoRegister(const char *name,
                const char *path,
                const char *cname,
                PetscErrorCode (*create) (Tao));

where name is the name of the solver (i.e., tao_blmvm), path is the path to the library containing the solver, cname is the name of the routine that creates the solver (in our case, TaoCreate_CG), and create is a pointer to that creation routine. If one is using dynamic loading, then the fourth argument will be ignored.

Once the solver has been registered, the new solver can be selected either by using the TaoSetType() function or by using the -tao_type command line option.

Footnotes

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For more on MPI and PETSc, see Running PETSc Programs.

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