Relaxation schemes for mathematical programs with switching constraints

TL;DR

This paper adapts a relaxation method to solve switching-constrained optimization problems, which are challenging due to their disconnected feasible sets and failure of standard qualifications, demonstrating convergence to M-stationary points.

Contribution

It introduces an adapted relaxation scheme for switching-constrained problems that reliably computes M-stationary points, unlike other schemes that only find W-stationary points.

Findings

The proposed method computes M-stationary points under mild assumptions.

Other relaxation schemes tend to find only W-stationary points.

Computational experiments show the effectiveness of the proposed relaxation method.

Abstract

Switching-constrained optimization problems form a difficult class of mathematical programs since their feasible set is almost disconnected while standard constraint qualifications are likely to fail at several feasible points. That is why the application of standard methods from nonlinear programming does not seem to be promising in order to solve such problems. In this paper, we adapt the relaxation method from Kanzow and Schwartz (SIAM J. Optim., 23(2):770-798, 2013) for the numerical treatment of mathematical programs with complementarity constraints to the setting of switching-constrained optimization. It is shown that the proposed method computes M-stationary points under mild assumptions. Furthermore, we comment on other possible relaxation approaches which can be used to tackle mathematical programs with switching constraints. As it turns out, adapted versions of Scholtes' global relaxation scheme as well as the relaxation scheme of Steffensen and Ulbrich only find W-stationary points of switching-constrained optimization problems in general. Some computational experiments visualize the performance of the proposed relaxation method.