Variable Aggregation-based Perspective Reformulation for Mixed-Integer Convex Optimization with Symmetry

TL;DR

This paper introduces a novel reformulation technique combining perspective reformulation with variable aggregation to tighten the continuous relaxation of symmetric mixed-integer convex problems, improving solution quality.

Contribution

It presents a new formulation integrating perspective reformulation into variable aggregation, providing exact convex hull characterizations for symmetric mixed-integer convex problems.

Findings

The proposed reformulation yields a tighter continuous relaxation.

Convex hulls of aggregated feasible regions are exactly characterized.

Theoretical foundations improve understanding of symmetry in mixed-integer convex optimization.

Abstract

This paper addresses the challenging issue of symmetry in mixed-integer convex optimization problems, which frequently arise in real-world applications such as the unit commitment problem. Although variable aggregation techniques have been employed to mitigate symmetry, their impact on tightening the corresponding continuous relaxation has not been thoroughly investigated. In this work, we propose a new formulation that integrates the perspective reformulation method into the variable aggregation framework, yielding a tighter continuous relaxation for mixed-integer convex optimization problems with symmetric structures. We prove that, in the presence of symmetry, the convex hull of the feasible region associated with each set of aggregated variables can be exactly characterized. These results demonstrate the effectiveness of the proposed reformulation and establish new theoretical foundations for achieving tightness in variable aggregation-based mixed-integer programming formulations.