Coordinate Optimality Reformulation for Mixed-Integer Convex Programs with Indicators

2026-08-02

AI Summary

This work addresses the computational inefficiency in solving mixed-integer convex optimization problems involving binary indicator variables that govern continuous variables. To tackle this challenge, the authors propose the Coordinate Optimality Reconstruction (CORe) framework, which uniquely integrates coordinate-wise optimality conditions into the modeling of indicator variables. By combining closed-form characterizations with disjunctive reformulation techniques, CORe constructs a novel mixed-integer convex programming formulation that effectively exploits exploitable structures embedded in the problem’s sparsity pattern. The approach preserves global optimality while substantially enhancing the performance of branch-and-bound algorithms. Experimental results demonstrate that, across multiple problem classes—including quadratic programs and robust single-index models—CORe significantly accelerates solver convergence compared to conventional big-M formulations.

Abstract

We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the \emph{Coordinate Optimality Reformulation} (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable problem structure. We first develop the main components of CORe, including coordinate-wise optimality conditions, closed-form characterizations, and disjunctive reformulations. We then demonstrate the framework across multiple problem families, including quadratic problems and robust single-index models. Computational experiments show that CORe can substantially improve solver performance compared with standard big-$M$ formulations.

Problem

• mixed-integer convex optimization
• indicator variables
• branch-and-bound
• problem structure
• optimality conditions

Innovation

• Coordinate Optimality Reformulation
• Mixed-Integer Convex Programming
• Indicator Variables
• Branch-and-Bound
• Disjunctive Reformulation