Computing Binary Integer Programming via A New Exact Penalty Function
Shuai Li, Shenglong Zhou
TL;DR
This work tackles unconstrained binary integer programming (UBIP) by reformulating binary constraints as continuous equalities on the unit box via a piecewise cubic penalty, enabling an exact penalty approach with a solution-independent threshold. It introduces P-stationary points and develops the Adaptive Proximal Point Algorithm (APPA), which provably converges to a P-stationary point and terminates in finite steps under mild assumptions. Theoretical contributions include a penalty-theorem linking the original and penalized problems without dependence on the unknown solution set, and convergence guarantees for APPA under strong smoothness/convexity. Empirical results across recovery, MIMO detection, and QUBO demonstrate that APPA achieves higher accuracy and substantially faster runtimes than state-of-the-art solvers, including GUROBI, highlighting its practical impact for large-scale binary optimization tasks.
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.