A constraint dissolving inexact penalty method for optimization problems with geometric constraints

Abstract

Optimization problems with geometric constraints have a broad range of applications, including machine learning, finance, and control. A powerful algorithmic tool to resolve these geometric constraints are constraint dissolving methods. To this end, we propose a framework for constraint dissolving mappings for nonconvex geometric constraints. Leveraging these, we develop a constraint dissolving inexact penalty method to solve optimization problems with general set-membership constraints and possibly nonconvex geometric constraints. We establish the convergence of the proposed algorithm and prove that every feasible accumulation point is Mordukhovich stationary. Notably, we rely only on mild asymptotic Mordukhovich regularity, which is significantly weaker than the constraint qualifications adopted in the existing literature on constraint dissolving methods. Numerical experiments addressing classical equality-, complementarity-, sparsity-, and low-rank constrained optimization problems demonstrate that the proposed method is competitive with the safeguarded augmented Lagrangian method in terms of solution quality and significantly outperforms the penalty decomposition method.