Tao (Optimization) - Low-level Interface

The Tao (Toolkit for Advanced Optimization) component provides methods for solving optimization problems, including unconstrained minimization, bound-constrained optimization, and constrained optimization.

Overview

Tao supports various optimization algorithms:

Unconstrained: Conjugate gradient (CG), limited-memory BFGS (BLMVM), Nelder-Mead
Bound-constrained: Bound-constrained BFGS (BNCG, BNLS, BNTL), active-set methods
Constrained: Augmented Lagrangian methods (ALMM), interior point methods
Least-squares: Gauss-Newton methods, Levenberg-Marquardt
Complementarity: Semismooth methods for complementarity problems
PDE-constrained: Methods suitable for PDE-constrained optimization

The Tao object manages the optimization process, line searches, convergence monitoring, and provides a unified interface across different optimization algorithms.

Basic Usage

using PETSc, MPI

# Initialize MPI and PETSc
MPI.Init()
petsclib = PETSc.getlib()
PETSc.initialize(petsclib)

# Create a Tao object
tao = LibPETSc.TaoCreate(petsclib, LibPETSc.PETSC_COMM_SELF)

# Set the optimization algorithm (e.g., LMVM, BLMVM, NLS)
LibPETSc.TaoSetType(petsclib, tao, "lmvm")

# Set the objective function and gradient
# LibPETSc.TaoSetObjective(petsclib, tao, objective_function_ptr, C_NULL)
# LibPETSc.TaoSetGradient(petsclib, tao, C_NULL, gradient_function_ptr, C_NULL)

# For bound-constrained problems, set variable bounds
# LibPETSc.TaoSetVariableBounds(petsclib, tao, lower_bound_vec, upper_bound_vec)

# Set convergence tolerances
LibPETSc.TaoSetTolerances(petsclib, tao, 1e-8, 1e-8, 1e-8)

# Set maximum iterations
LibPETSc.TaoSetMaximumIterations(petsclib, tao, 1000)

# Set options from command line/options database
LibPETSc.TaoSetFromOptions(petsclib, tao)

# Set initial guess
# LibPETSc.TaoSetSolution(petsclib, tao, initial_vec)

# Solve the optimization problem
# LibPETSc.TaoSolve(petsclib, tao)

# Get solution information
reason = LibPETSc.TaoGetConvergedReason(petsclib, tao)
iter = LibPETSc.TaoGetIterationNumber(petsclib, tao)

println("Tao reason: $(reason), iterations: $(iter)")

# Cleanup
LibPETSc.TaoDestroy(petsclib, tao)

# Finalize PETSc and MPI
PETSc.finalize(petsclib)
MPI.Finalize()

Common Workflow

Create and configure Tao: Use TaoCreate, TaoSetType
Define objective: TaoSetObjective, TaoSetGradient, optionally TaoSetHessian
Set constraints (if any): TaoSetVariableBounds, TaoSetConstraints
Configure solver: TaoSetTolerances, TaoSetMaximumIterations
Set initial guess: TaoSetSolution
Solve: TaoSolve
Retrieve solution: TaoGetSolution, TaoGetConvergedReason

Optimization Algorithms

Available through TaoSetType:

Unconstrained:
- TAOLMVM: Limited-memory variable metric (quasi-Newton)
- TAOCG: Conjugate gradient methods
- TAONM: Nelder-Mead simplex method
- TAONLS: Newton line search
- TAONTL: Newton trust-region with line search
Bound-constrained:
- TAOBLMVM: Bound-constrained limited-memory variable metric
- TAOBNCG: Bound-constrained conjugate gradient
- TAOBQNLS: Bound-constrained quasi-Newton line search
- TAOBNTL: Bound-constrained Newton trust-region
- TAOTRON: Trust-region Newton method

The types built on an LMVM matrix (lmvm, blmvm, bncg, bqnls, bqnkls, bntl and bntr) fail after a PETSc.finalize/PETSc.initialize cycle in which a KSP was created: PETSc 3.25 forgets the LMVM matrix types at PetscFinalize and has no public way to register them again, so TaoSetType throws with "Unknown Mat type given: lmvmbfgs". Initialize once per Julia session, or restart Julia. cg, nm, nls, ntr, ntl and tron are not affected.

Constrained:
- TAOALMM: Augmented Lagrangian multiplier method
- TAOIPM: Interior point method
- TAOPDIPM: Primal-dual interior point method
Least-squares:
- TAOPOUNDERS: POUNDERs model-based method
- TAOBRGN: Bounded regularized Gauss-Newton
Complementarity:
- TAOSSLS: Semismooth least squares
- TAOASLS: Active-set least squares

Convergence Criteria

Tao monitors several convergence criteria:

Gradient tolerance: ||∇f|| < gatol or ||∇f||/||f|| < grtol
Function tolerance: |f - f_prev| < fatol or |f - f_prev|/|f| < frtol
Step tolerance: ||x - x_prev|| < steptol

Set using TaoSetTolerances(tao, gatol, grtol, gttol).

Hessian Options

For second-order methods:

Exact Hessian: Provide via TaoSetHessian
Finite-difference approximation: Use matrix-free approach
Quasi-Newton approximation: LMVM methods build approximation automatically

Function Reference

PETSc.LibPETSc.TaoADMMGetDualVector — Method

Y::PetscVec = TaoADMMGetDualVector(petsclib::PetscLibType, tao::AbstractTao)

Returns the dual vector associated with the current TAOADMM state

Not Collective

Input Parameter:

tao - the Tao context

Output Parameter:

Y - the current solution

Level: intermediate

See also: TAOADMM

External Links