Topological approach to mathematicalprograms with switching constraints

TL;DR

This paper applies topological methods, specifically Morse theory, to analyze the structure of mathematical programs with switching constraints, revealing how the topology of level sets changes at stationary points and establishing conditions for stability.

Contribution

It introduces a topological framework for MPSC, proves Morse-theoretic theorems, and characterizes nondegeneracy and stability of stationary points in this context.

Findings

Topology of level sets changes via cell attachment at W-stationary points

All W-stationary points are generically nondegenerate

Characterization of strong stability using first and second order conditions

Abstract

We study mathematical programs with switching constraints (MPSC)from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous defor-mation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian's Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy andstrong stability of W-stationary points. A complete characterizationof strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, all bi-active Lagrange multipliers of a strongly stable W-stationary point cannot vanish.