Computational Comparison of Exact Solution Methods for 0-1 Quadratic Programs: Recommendations for Practitioners

Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, second-order cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our framework, we re-derive previous relaxation schemes and provide new ones. In particular, we present three relaxations for binary quadratic polynomial programs. The first two relaxations, based on second-order cone and semidefinite programming, represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems. From a practical point of view, due to the computational cost, semidefinite-based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances. To improve the computational efficiency for solving such problems, we propose a third relaxation based purely on second-order cone programming. Computational tests on different classes of nonconvex binary quadratic polynomial problems, including quadratic knapsack problems, show that the second-order-cone-based relaxation outperforms the semidefinite-based relaxations that are proposed in the literature in terms of computational efficiency, and it is comparable in terms of bounds.

A linearization technique for binary quadratic programs (BQPs) that comprise linear constraints is presented. The technique, called “inductive linearization”, extends concepts for BQPs with particular equation constraints, that have been referred to as “compact linearization” before, to the general case. Quadratic terms may occur in the objective function, in the set of constraints, or in both. For several relevant applications, the linear programming relaxations obtained from applying the technique are proven to be at least as strong as the one obtained with a well-known classical linearization. It is also shown how to obtain an inductive linearization automatically. This might be used, e.g., by general-purpose mixed-integer programming solvers.

Genomic selection (GS) in plant breeding aims to identify individuals with superior genetic merit while maintaining genetic diversity within populations. In plant breeding, considering multiple traits simultaneously makes optimizing selection complex, especially under genetic relatedness constraints. In this study, we propose a binary quadratic programming framework for constructing a multi-trait selection index that maximizes genetic gain while minimizing average pairwise relatedness appropriate for identifying superior candidates for advancement in the breeding pipeline. The approach combines estimated breeding values (EBVs) across multiple traits by applying trait-specific economic weights, while simultaneously accounting for coancestry through the genomic relationship matrix. By formulating the selection problem as a constrained Quadratic Programing Multi-trait Selection Index (QPMSI), our method enables the identification of a fixed number of candidate individuals that jointly optimize selection index values and control genetic relatedness. We evaluated the performance of the proposed method using five real genomic datasets and demonstrated that it provides a more effective balance between selection response and control of genetic relatedness than the Linear Programming Multi-trait Selection Index (LPMSI). In particular, the QPMSI consistently outperformed the LPMSI in terms of the MV metric (gain-to-degree of relatedness ratio), achieving improvements of at least 53.8%. This framework offers a practical and computationally efficient tool for sustainable breeding strategies in multi-trait selection contexts.

We consider the problem of handling binary constraints in optimization problems. We review methods for solving binary quadratic programs (BQPs), such as the spectral method and semidefinite programming relaxations. We then discuss two new methods for handling these constraints. The first involves the introduction of an extra unconstrained variable along with the alternating direction method of multipliers (ADMM), which has now also appeared in other recent literature. This allows the effect of the binary variables to be decoupled when they are minimized over. The second involves rewarding the one-norm while restricting the infinity-norm, based on a reformulation of the original problem. The piecewise linearity of the negative penalty results in the problem being convex until it hits a critical point, at which point the parameters of this linear term can be changed. These two methods can be applied to any problems which are convex except for binary constraints. In addition to testing them on BQPs, we show the efficacy of these approaches on point segmentation and image segmentation problems.

The multiple-input multiple-output (MIMO) system is widely used in wireless communications. For the problem of the discrete maximum-likelihood (ML) detection for the MIMO system, one can formulate it as binary quadratic programming (BQP). The general BQP problem is an NP-hard problem, which is a challenge for finding promising solutions. The variable complexity is a special considered issue. In this paper, inspired by the optimization of sparse constrains, we employ a regularization approach to deal with the binary constraints in the proposed formulation and then introduce the difference of convex functions (DC) programming to solve the formulated nonconvex cost function. A novel and robust DC algorithm is proposed. Numerical experiments show that the proposed algorithm, which is based on DC programming, can achieve accurate results with a higher convergence rate.

Binary quadratic programming (BQP) is a typical integer programming problem widely applied in the field of signal processing, economy, management and engineering. However, it's NP-hard and lacks efficient algorithms. Due to this reason, in this paper, a novel polynomial algorithm to linearly constrained binary quadratic programming problems with Q being a five-diagonal matrix is focused by combining the basic algorithm proposed in [1], [2], [3] and the dynamic programming method. We first briefly deduce the basic algorithm. Then, the algorithm is proposed to solve this special problem. In addition, a specific example is presented to illustrate the new algorithm. Lastly, we demonstrate its polynomial feature as well as its high efficiency.

Epilepsy is one of the most common disorders of the nervous system. The progressive entrainment between an epileptogenic focus and normal brain areas results to transitions of the brain from chaotic to less chaotic spatiotemporal states, the epileptic seizures. The entrainment between two brain sites can be quantified by the T-index from the measures of chaos (e.g., Lyapunov exponents) of the electrical activity (EEG) of the brain. By applying the optimization theory, in particular quadratic zero-one programming, we were able to select the most entrained brain sites 10 minutes before seizures and subsequently follow their entrainment over 2 hours before seizures. In five patients with 3–24 seizures, we found that over 90% of the seizures are predictable by the optimal selection of electrode sites. This procedure, which is applied to epilepsy research for the first time, shows the possibility of prediction of epileptic seizures well in advance (19.8 to 42.9 minutes) of their occurrence.

A way of minimizing the opportunity of cheating in exams is to assign different tests to students. The likelihood of cheating then depends on the proximity of the students' desks, and the similarity of the tests. The test-assignment problem is to find an assignment of tests to desks that minimizes that total likelihood of cheating. The problem is a variant of a graph coloring problem and is NP-hard. We propose a new heuristic solution for this problem. Our approach differs from the usual way of designing heuristics in two ways. First, we reduce test-assignment to the more general unconstrained binary quadratic programming. Second, we search for a good heuristic using an automatic algorithm configuration tool that evolves heuristics in a space of algorithms built from known components for binary quadratic programming. The best hybrid heuristics found repeatedly recombine elements of a population of elite solutions and improve them by a tabu search. Computational tests suggest that the resulting algorithms are competitive with existing heuristics that have been designed manually.

Maximum a posteriori (MAP) inference in Markov random field (MRF) lays the foundation for many computer vision tasks, which can be formulated by a binary quadratic programming (BQP) problem. Compar...

The computational costs of Cauchy Reed-Solomon (CRS) encoding operation make a great impact on the performance of its practical applications. The letter concentrates on how to construct a good Cauchy matrix which can lead to an efficient CRS coding scheme. We first formally model the problem by using a binary quadratic programming, then present an approximate method called localized greedy algorithm (LGA) to solve it. Compared with existing work, LGA requires much lower complexities to obtain the same performance of Cauchy matrices.

Multi-Product Newsvendor Problem with Customer-Driven Demand Substitution: A Stochastic Integer Program Perspective

We review some of the recent enhancements of interior-point methods for the improved solution of semidefinite relaxations in combinatorial optimization and binary quadratic programming. Central topics include general interior-point cutting-plane schemes, handling of linear inequalities, and several warm-starting strategies. A practical implementation and computational results are also discussed.

This paper proposes a generic formulation that significantly expedites the training and deployment of image classification models, particularly under the scenarios of many image categories and high feature dimensions. As the core idea, our method represents both the images and learned classifiers using binary hash codes, which are simultaneously learned from the training data. Classifying an image thereby reduces to retrieving its nearest class codes in the Hamming space. Specifically, we formulate multiclass image classification as an optimization problem over binary variables. The optimization alternatingly proceeds over the binary classifiers and image hash codes. Profiting from the special property of binary codes, we show that the sub-problems can be efficiently solved through either a binary quadratic program (BQP) or a linear program. In particular, for attacking the BQP problem, we propose a novel bit-flipping procedure which enjoys high efficacy and a local optimality guarantee. Our formulation supports a large family of empirical loss functions and is, in specific, instantiated by exponential and linear losses. Comprehensive evaluations are conducted on several representative image benchmarks. The experiments consistently exhibit reduced computational and memory complexities of model training and deployment, without sacrificing classification accuracy.

Federated‐cloud has been widely deployed due to the growing popularity of real‐time applications, and hence allocating resources among clouds becomes nontrivial to meet the stringent service requirements. The challenges lie in achieving minimized latency constrained by virtual machines rental overhead and resource requirement. This becomes further complicated by the issues of datacenter selection. To this end, we propose AIMING, a novel resource allocation approach which aims to minimize the latency constrained by monetary overhead in the context of federated‐cloud. Specifically, the network resources are deployed and selected according to k‐means clustering. Meanwhile, the total latency among datacenters is optimized based on binary quadratic programming. The evaluation is conducted with real data traces. The results show that AIMING can reduce total datacenter latency effectively compared with other approaches.

This paper presents a novel halftoning algorithm based on continuous relaxation and gradient-based optimization. By reformulating the original binary quadratic programming (BQP) problem into a differentiable non-convex objective, the proposed method integrates a perceptual loss with a binarization regularizer to guide the optimization toward high-quality binary outputs. Unlike deep learning-based approaches, our method is model-free, memory-efficient, and fully interpretable, enabling fast convergence without complex training or large memory overhead. Experimental results demonstrate that the proposed method outperforms both classical and neural halftoning techniques in terms of image quality and convergence speed. This work provides a practical and scalable solution for highresolution halftone image generation, particularly suitable for industrial and real-time applications under resource constraints.