AugmentedLagrangianOptimizer

Overview¶

AugmentedLagrangianOptimizer combines penalty terms with Lagrange multipliers to improve convergence and conditioning on constrained nonlinear problems.

Authors: Frank Dellaert, et al. (see THANKS for the full author list)

See LICENSE for the license information

try:
    import google.colab
    %pip install --quiet gtsam-develop
except ImportError:
    pass

Key Concepts¶

AugmentedLagrangianState extends penalty state with equality and inequality multipliers.
augmentedLagrangianFunction builds the constrained objective used in each outer iteration.
Multiplier updates are done by dual-ascent style steps.
Penalty parameters are adapted based on violation reduction.

Mathematical Formulation¶

Augmented Lagrangian combines multipliers and penalties:

\mathcal{L}_A(x,\lambda)=\frac{1}{2}\|f(x)\|^2 -\lambda_{eq}^T h(x)+\frac{\mu_{eq}}{2}\|h(x)\|^2 -\lambda_{ineq}^T g(x)+\frac{\mu_{ineq}}{2}\|g(x)_-\|^2.

(1)

The solver alternates between primal minimization (in $x$ ) and dual-ascent updates (in multipliers), with adaptive penalty updates.

Key User API¶

AugmentedLagrangianOptimizer(problem, initialValues, params)
optimize()
progress()
augmentedLagrangianFunction(state, epsilon) (advanced inspection)
AugmentedLagrangianParams: dual step sizes and penalty-adaptation settings

Concise C++ Example¶

#include <gtsam/constrained/AugmentedLagrangianOptimizer.h>

using namespace gtsam;

auto params = std::make_shared<AugmentedLagrangianParams>();
params->verbose = true;

AugmentedLagrangianOptimizer optimizer(problem, init_values, params);
Values results = optimizer.optimize();