TAOALMM#

Augmented Lagrangian multiplier method for solving nonlinear optimization problems with general constraints.

Options Database Keys#

-tao_almm_mu_init - initial penalty parameter (default: 10.)
-tao_almm_mu_factor - increase factor for the penalty parameter (default: 100.)
-tao_almm_mu_max - maximum safeguard for penalty parameter updates (default: 1.e20)
-tao_almm_mu_power_good - exponential for penalty parameter when multiplier update is accepted (default: 0.9)
-tao_almm_mu_power_bad - exponential for penalty parameter when multiplier update is rejected (default: 0.1)
-tao_almm_ye_min - minimum safeguard for equality multiplier updates (default: -1.e20)
-tao_almm_ye_max - maximum safeguard for equality multiplier updates (default: 1.e20)
-tao_almm_yi_min - minimum safeguard for inequality multiplier updates (default: -1.e20)
-tao_almm_yi_max - maximum safeguard for inequality multiplier updates (default: 1.e20)
-tao_almm_type <phr,classic> - change formulation of the augmented Lagrangian merit function for the subproblem (default: phr)

Notes#

This method converts a constrained problem into a sequence of unconstrained problems via the augmented Lagrangian merit function. Bound constraints are pushed down to the subproblem without any modifications.

Two formulations are offered for the subproblem: canonical Hestenes-Powell augmented Lagrangian with slack variables for inequality constraints, and a slack-less Powell-Hestenes-Rockafellar (PHR) formulation utilizing a pointwise max() penalty on inequality constraints. The canonical augmented Lagrangian formulation may converge faster for smaller problems but is highly susceptible to poor step lengths in the subproblem due to the positivity constraint on slack variables. PHR avoids this issue by eliminating the slack variables entirely, and is highly desirable for problems with a large number of inequality constraints.

The subproblem is solved using a nested first-order TAO solver (default: TAOBQNLS). The user can retrieve a pointer to the subsolver via TaoALMMGetSubsolver() or pass command line arguments to it using the “-tao_almm_subsolver_” prefix. Currently, TAOALMM does not support second-order methods for the subproblem.

  while unconverged
    solve argmin_x L(x) s.t. l <= x <= u
    if ||c|| <= y_tol
      if ||c|| <= c_tol && ||Lgrad|| <= g_tol:
        problem converged, return solution
      else
        constraints sufficiently improved
        update multipliers and tighten tolerances
      endif
    else
      constraints did not improve
      update penalty and loosen tolerances
    endif
  endwhile