A Study of One-Parameter Regularization Methods for Mathematical Programs with Vanishing Constraints

TL;DR

This paper compares four regularization methods for solving mathematical programs with vanishing constraints, providing theoretical convergence analysis and numerical experiments demonstrating their effectiveness in engineering applications.

Contribution

It introduces and analyzes four regularization techniques for MPVCs, establishing convergence properties and identifying promising schemes through numerical experiments.

Findings

Regularization improves solution stability over standard solvers.

Two schemes show consistent convergence and efficiency.

Numerical results validate theoretical convergence and practical benefits.

Abstract

Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often prevents standard constraint qualifications and optimality conditions from being attained; moreover, the feasible set is inherently nonconvex, and often has no interior around points of interest. In this paper, we therefore study and compare four regularization methods for the numerical solution of MPVCS. Each method depends on a single regularization parameter, which is used to embed the original MPVC into a sequence of standard nonlinear programs. Convergence results for these methods based on both exact and approximate stationary of the subproblems are established under weak assumptions. The improved regularity of the subproblems is studied by providing sufficient conditions for the existence of KKT multipliers. Numerical experiments, based on applications in truss topology design and an optimal control problem from aerothermodynamics, complement the theoretical analysis and comparison of the regularization methods. The computational results highlight the benefit of using regularization over applying a standard solver directly, and they allow us to identify two promising regularization schemes.