Computing Binary Integer Programming via A New Exact Penalty Function

TL;DR

This paper introduces a novel reformulation and an exact penalty framework for unconstrained binary integer programming, enabling efficient algorithms with guaranteed convergence and improved performance over existing solvers.

Contribution

A new reformulation using a piecewise cubic function and an exact penalty approach with a solution-independent threshold, along with the development of the APPA algorithm for UBIP.

Findings

APPA converges to a P-stationary point within finite iterations

APPA outperforms existing solvers in accuracy and efficiency

The penalty threshold is independent of the unknown solution set

Abstract

Unconstrained binary integer programming (UBIP) poses significant computational challenges due to its discrete nature. We introduce a novel reformulation approach using a piecewise cubic function that transforms binary constraints into continuous equality constraints. Instead of solving the resulting constrained problem directly, we develop an exact penalty framework with a key theoretical advantage: the penalty parameter threshold ensuring exact equivalence is independent of the unknown solution set, unlike classical exact penalty theory. To facilitate the analysis of the penalty model, we introduce the concept of P-stationary points and systematically characterize their optimality properties and relationships with local and global minimizers. The P-stationary point enables the development of an efficient algorithm called APPA, which is guaranteed to converge to a P-stationary point within a finite number of iterations under a single mild assumption, namely, strong smoothness of the objective function over the unit box. Comprehensive numerical experiments demonstrate that APPA outperforms established solvers in both accuracy and efficiency across diverse problem instances.