Inexact Penalty Decomposition Methods for Optimization Problems with Geometric Constraints

TL;DR

This paper introduces a penalty decomposition method for complex geometric constrained optimization problems, effectively handling nonconvex constraints and demonstrating superior numerical performance over existing approaches.

Contribution

It presents a novel inexact penalty decomposition scheme that explicitly manages difficult constraints, generalizing previous methods and improving numerical efficiency.

Findings

Method effectively handles nonconvex and complicated constraints.

Numerical results show high efficiency on complex vector and matrix problems.

Decomposition approach offers more flexibility and fewer projection steps.

Abstract

This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and supposingly simple) constraints are tackled by sequential penalization. Inexact optimization steps are proven sufficient for the esulting algorithm to work, so that it is employable even with difficult objective functions. The current work is therefore a significant generalization of existing papers on penalty decomposition methods. On the other hand, it is related to some recent publications which use an augmented Lagrangian idea to solve optimization problems with geometric constraints. Compared to these methods, the decomposition idea is shown to be numerically superior since it allows much more freedom in the choice of the subproblem solver, and since the number of certain (possibly expensive) projection steps is significantly less. Extensive numerical results on several highly complicated classes of optimization problems in vector and matrix paces indicate that the current method is indeed very efficient to solve these problems.