petsc-main 2021-04-20
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TaoALMM

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

Options Database Keys

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

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 typically converges faster for most problems. However, PHR may be desirable for problems featuring a large number of inequality constraints because it avoids inflating the size of the subproblem with slack variables.

The subproblem is solved using a nested first-order TAO solver. 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. It is also highly recommended that the subsolver chosen by the user utilize a trust-region strategy for globalization (default: TAOBQNKTR) especially if the outer problem features bound constraints.

  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

See Also

TaoALMMGetType(), TaoALMMSetType(), TaoALMMSetSubsolver(), TaoALMMGetSubsolver(),
TaoALMMGetMultipliers(), TaoALMMSetMultipliers(), TaoALMMGetPrimalIS(), TaoALMMGetDualIS()

Level

beginner

Location

src/tao/constrained/impls/almm/almm.c
Index of all Tao routines
Table of Contents for all manual pages
Index of all manual pages