1: /* Context for SSXLS
  2:    -- semismooth (SS) - function not differentiable
  3:                       - merit function continuously differentiable
  4:                       - Fischer-Burmeister reformulation of complementarity
  5:                         - Billups composition for two finite bounds
  6:    -- infeasible (I)  - iterates not guaranteed to remain within bounds
  7:    -- feasible (F)    - iterates guaranteed to remain within bounds
  8:    -- linesearch (LS) - Armijo rule on direction

 10:  Many other reformulations are possible and combinations of
 11:  feasible/infeasible and linesearch/trust region are possible.

 13:  Basic theory
 14:    Fischer-Burmeister reformulation is semismooth with a continuously
 15:    differentiable merit function and strongly semismooth if the F has
 16:    lipschitz continuous derivatives.

 18:    Every accumulation point generated by the algorithm is a stationary
 19:    point for the merit function.  Stationary points of the merit function
 20:    are solutions of the complementarity problem if
 21:      a.  the stationary point has a BD-regular subdifferential, or
 22:      b.  the Schur complement F'/F'_ff is a P_0-matrix where ff is the
 23:          index set corresponding to the free variables.

 25:    If one of the accumulation points has a BD-regular subdifferential then
 26:      a.  the entire sequence converges to this accumulation point at
 27:          a local q-superlinear rate
 28:      b.  if in addition the reformulation is strongly semismooth near
 29:          this accumulation point, then the algorithm converges at a
 30:          local q-quadratic rate.

 32:  The theory for the feasible version follows from the feasible descent
 33:  algorithm framework.

 35:  References:
 36:    Billups, "Algorithms for Complementarity Problems and Generalized
 37:      Equations," Ph.D thesis, University of Wisconsin - Madison, 1995.
 38:    De Luca, Facchinei, Kanzow, "A Semismooth Equation Approach to the
 39:      Solution of Nonlinear Complementarity Problems," Mathematical
 40:      Programming, 75, pages 407-439, 1996.
 41:    Ferris, Kanzow, Munson, "Feasible Descent Algorithms for Mixed
 42:      Complementarity Problems," Mathematical Programming, 86,
 43:      pages 475-497, 1999.
 44:    Fischer, "A Special Newton-type Optimization Method," Optimization,
 45:      24, pages 269-284, 1992
 46:    Munson, Facchinei, Ferris, Fischer, Kanzow, "The Semismooth Algorithm
 47:      for Large Scale Complementarity Problems," Technical Report 99-06,
 48:      University of Wisconsin - Madison, 1999.
 49: */

 51: #ifndef __TAO_SSLS_H
 53: #include <petsc/private/taoimpl.h>

 55: typedef struct {
 56:   Vec ff;       /* fischer function */
 57:   Vec dpsi;     /* gradient of psi */

 59:   Vec da;       /* work vector for subdifferential calculation (diag pert) */
 60:   Vec db;       /* work vector for subdifferential calculation (row scale) */
 61:   Vec dm;   /* work vector for subdifferential calculation (mu vector) */
 62:   Vec dxfree;

 64:   Vec t1;       /* work vector */
 65:   Vec t2;       /* work vector */

 67:   Vec r1,r2,r3,w; /* work vectors */

 69:   PetscReal merit; /* merit function value (norm(fischer)) */
 70:   PetscReal merit_eqn;
 71:   PetscReal merit_mu;

 73:   PetscReal delta;
 74:   PetscReal rho;

 76:   PetscReal rtol;       /* Solution tolerances */
 77:   PetscReal atol;

 79:   PetscReal identifier; /* Active-set identification */

 81:   /* Interior-point method data */
 82:   PetscReal mu_init; /* initial smoothing parameter value */
 83:   PetscReal mu;      /* smoothing parameter */
 84:   PetscReal dmu;     /* direction in smoothing parameter */
 85:   PetscReal mucon;   /* smoothing parameter constraint */
 86:   PetscReal d_mucon; /* derivative of smoothing constraint with respect to mu */
 87:   PetscReal g_mucon; /* gradient of merit function with respect to mu */

 89:   Mat J_sub, Jpre_sub; /* subset of jacobian */
 90:   Vec f;        /* constraint function */

 92:   IS fixed;
 93:   IS free;
 94: } TAO_SSLS;

 96: PetscErrorCode TaoSetFromOptions_SSLS(PetscOptionItems *,Tao);
 97: PetscErrorCode TaoView_SSLS(Tao,PetscViewer);

 99: PetscErrorCode Tao_SSLS_Function(TaoLineSearch, Vec, PetscReal *, void *);
100: PetscErrorCode Tao_SSLS_FunctionGradient(TaoLineSearch, Vec, PetscReal *, Vec, void *);

102: #endif