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arXiv:2606.28734

Local Minima in Quadratic-Penalty Relaxations of Binary Linear Programs

Cheng-Han Huang, Yongliang Sun, Chaoyan Huang, Ismail Alkhouri, Rongrong Wang

Published Jun 27, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2606.28734

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Quadratic unconstrained binary optimisation (QUBO) problems is a general class of combinatorial optimisation problems, which regained popularity after recent advances in quantum computing. Quantum-inspired technologies like Fujitsu’s Digital Annealer (DA), based on simulated annealing, can solve QUBO problems much faster than traditional computers. Penalty methods can convert constrained optimisation problems into QUBO problems. However, existing exact methods of determining penalty weights are limited to specific QUBO problems and require manual analysis by experts in QUBO. Empirical methods are general but become computationally prohibitive for large problem sizes and do not guarantee that penalty weights preserve the feasibility of global optima. We present a simple, efficient, general method applicable to any QUBO to determine exact penalty weights. Such weights are simple upper-bounds of the smallest penalty weight guaranteeing that unconstrained global optima are the same as the feasible global optima. These bounds can be iteratively improved by sequential penalty methods which we also present. Experimental results with the DA on minimum cut, travelling salesman and multi-dimensional knapsack problems show the viability of the novel methodology hybridising exact and sequential methods. This work contributes towards general, automatic and scalable penalty methods in QUBO.
Quadratic unconstrained binary optimisation (QUBO) problems is a general class of combinatorial optimisation problems, which regained popularity after recent advances in quantum computing. Quantum-inspired technologies like Fujitsu's Digital Annealer (DA), based on simulated annealing, can solve QUBO problems much faster than traditional computers. Penalty methods can convert constrained optimisation problems into QUBO problems. However, existing exact methods of determining penalty weights are limited to specific QUBO problems and require manual analysis by experts in QUBO. Empirical methods are general but become computationally prohibitive for large problem sizes and do not guarantee that penalty weights preserve the feasibility of global optima. We present a simple, efficient, general method applicable to any QUBO to determine exact penalty weights. Such weights are simple upper-bounds of the smallest penalty weight guaranteeing that unconstrained global optima are the same as the feasible global optima. These bounds can be iteratively improved by sequential penalty methods which we also present. Experimental results with the DA on minimum cut, travelling salesman and multi-dimensional knapsack problems show the viability of the novel methodology hybridising exact and sequential methods. This work contributes towards general, automatic and scalable penalty methods in QUBO.
Quantum computing has emerged as a promising alternative for solving combinatorial optimization problems. The standard approach for encoding optimization problems on quantum processing units (QPUs) involves transforming them into their Quadratic Unconstrained Binary Optimization (QUBO) representation. However, encoding constraints of optimization problems, particularly inequality constraints, into QUBO requires additional variables, which results in more qubits. Considering the limited availability of qubits in NISQ machines, existing encoding methods fail to scale due to their reliance on large numbers of qubits. We propose a generalized exponential penalty framework for QUBO inequality constraints inspired by a class of exponential functions, which we call exponential penalization. This paper presents an encoding strategy for inequality constraints in combinatorial optimization problems, inspired by a class of exponential functions, which we call exponential penalization. The initial idea of using exponential penalties for QUBO formulation was introduced by Montanez-Barrera et al. by applying a specific exponential function to reduce qubit requirements. In this work, we extend that approach by conducting a comprehensive study on a broader class of exponential functions, analyzing their theoretical properties and empirical performance. Our experimental results demonstrate that an exponential penalization achieves 57%, 83% qubit number reduction for Bin Packing Problem (BPP) and Traveling Salesman Problem (TSP), respectively. And we demonstrate comparable solution quality to classical with a probability of 6% and 21% accuracy for BPP with 8 and TSP with 12 qubits, respectively.
Many combinatorial optimization problems can be formulated as a problem to determine the order of sequence or to find a corresponding mapping of the objects. We call such problems permutation-based optimization problems. Many such problems can be formulated as a quadratic unconstrained binary optimization (QUBO) or Ising model by introducing a penalty coefficient to the permutation constraint terms. While classical and quantum annealing approaches have been proposed to solve QUBOs to date, they face issues with optimality and feasibility. Here we treat a given QUBO solver as a black box and propose techniques to enhance its performance. First, to ensure an effective search for good quality solutions, a smooth energy landscape is needed; we propose a data scaling approach that reduces the amplitudes of the input without compromising optimality. Second, we need to tune the penalty coefficient. In this paper, we illustrate that for certain problems, it suffices to tune the parameter by performing random sampling on the penalty coefficients to achieve good performance. Finally, to handle possible infeasibility of the solution, we introduce a polynomial-time projection algorithm. We apply these techniques along with a divide-and-conquer strategy to solve some large-scale permutation-based problems and present results for TSP and QAP.