Variable Aggregation-based Perspective Reformulation for Mixed-Integer Convex Optimization with Symmetry
Junhao Wu, Shaoze Li, Cheng Lu, Zhibin Deng, Shu-Cherng Fang
TL;DR
The paper tackles symmetry in mixed-integer convex optimization by integrating perspective reformulation with variable aggregation to tighten the continuous relaxation. It proves a convex-hull characterization for aggregated variable sets, showing that the continuous relaxation of the aggregated reformulation exactly captures the hull under mild conditions. The aggregated reformulation (P-agg) achieves relaxation bounds that match perspective-only formulations (P-per) and are strictly tighter than the original MICOP, while also effectively breaking symmetry. Computational experiments on unit commitment, line cover, and separable convex quadratic problems demonstrate substantial runtime gains and tighter lower bounds, highlighting the practical impact of the method for large, symmetric MICOP instances.
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.