The Exactness of the $\ell_1$ Penalty Function for a Class of Mathematical Programs with Generalized Complementarity Constraints

TL;DR

This paper investigates the conditions under which the $ ext{l}_1$ penalty function exactly captures solutions of a class of Mathematical Programs with Generalized Complementarity Constraints, including applications in physics and transportation.

Contribution

It establishes new exactness results for the $ ext{l}_1$ penalty function in MPGCC with multi-affine objectives, extending previous work and covering multi-block cases.

Findings

Provided an example where existing tools fail to establish exactness.

Proved exactness under mild conditions for a broad class of MPGCC.

Extended results to multi-block contexts and applications.

Abstract

In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relationships are imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The $\ell_1$ penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the $\ell_1$ penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problem in network transportation. We first provide an instance from this class, the exactness of whose $\ell_1$ penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.