Local Minima in Quadratic-Penalty Relaxations of Binary Linear Programs

2026-06-27

AI Summary

This work addresses the issue that quadratic penalty relaxations of binary linear programs often yield spurious or infeasible local minima. To overcome this, we propose a class of QUBO relaxation models satisfying specific structural conditions that guarantee all local minima are feasible and strictly binary. Leveraging these conditions, we derive novel differentiable relaxations for classical combinatorial optimization problems—including open-pit mining, the 0–1 knapsack problem, and the traveling salesman problem—and solve them using gradient-based optimizers such as projected gradient descent and Adam. Experimental results demonstrate that the proposed approach reliably converges to valid binary solutions, thereby establishing clear theoretical guarantees and delineating the applicability boundaries of differentiable optimization as a local solver for combinatorial problems.

Abstract

Many combinatorial optimization problems admit quadratic unconstrained binary formulations (QUBO) which can often be relaxed to the box $[0,1]^n$ and optimized using scalable gradient-based methods. However, the resulting non-convex landscape can often contain local optima that are spurious or infeasible. In this paper, we establish sufficient structural conditions on quadratic penalties that rule out these failures, guaranteeing that every local minimizer of the relaxed problem is both binary and feasible. For each problem we study, we examine existing QUBO formulations when available, identify why they fail when they do, and propose alternative relaxed QUBOs that satisfy our conditions. We show for several common combinatorial problems, including open-pit mining, 0--1 knapsack, and traveling salesman formulations, that these constructions allow gradient-based methods such as projected gradient descent and Adam to be safely applied to obtain valid binary solutions. Our results clarify when differentiable optimization is a reliable local solver for quadratic combinatorial objectives.

Problem

• local minima
• quadratic penalty
• binary linear programs
• QUBO
• non-convex optimization

Innovation

• quadratic penalty
• binary feasibility
• local minima
• QUBO relaxation
• gradient-based optimization