Compact extended formulations for low-rank functions with indicator variables

TL;DR

This paper introduces compact extended formulations for the convex hulls of low-rank functions with indicator variables, improving computational efficiency in mixed-integer convex optimization problems.

Contribution

It presents a novel disjunctive representation that yields polynomial-sized formulations in the number of variables, generalizing known results and enhancing solver performance.

Findings

Formulations are exponentially smaller than traditional disjunctive approaches.

The method generalizes existing convex hull results for low-dimensional cases.

Computational experiments show significant solver performance improvements.

Abstract

We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming, although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function, but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems.