An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

TL;DR

This paper introduces a specialized augmented Lagrangian method for solving complex optimization problems with geometric constraints, including nonconvex and disjunctive constraints, demonstrating its effectiveness through extensive numerical experiments.

Contribution

The paper develops a novel augmented Lagrangian approach that explicitly handles complicated geometric constraints and solves subproblems with a tailored projected gradient method, with proven convergence to stationary points.

Findings

Effective in solving nonconvex geometric constraints

Converges to Mordukhovich-stationary points under mild conditions

Shows strong numerical performance on complementarity, cardinality, and MAXCUT problems

Abstract

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.