Inductive linearization for binary quadratic programs with linear constraints

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.

This paper is concerned with binary quadratic programs (BQPs), which are among the most well-studied classes of nonlinear integer optimization problems because of their wide variety of applications. While a number of different solution approaches have been proposed for tackling BQPs, practitioners need techniques that are both efficient and easy to implement. We revisit two of the most widely used linearization strategies for BQPs and examine the effectiveness of enhancements to these formulations that have been suggested in the literature. We perform a detailed large-scale computational study over five different classes of BQPs to compare these two linearizations with a more recent linear reformulation and direct submission of the nonlinear integer program to an optimization solver. The goal is to provide practitioners with guidance on how to best approach solving BQPs in an effective and easily implemented manner.

We address the exact solution of general integer quadratic programs with linear constraints. These programs constitute a particular case of mixed-integer quadratic programs for which we introduce in Billionnet et al. (Math. Program., 2010) a general solution method based on quadratic convex reformulation, that we called MIQCR. This reformulation consists in designing an equivalent quadratic program with a convex objective function. The problem reformulated by MIQCR has a relatively important size that penalizes its solution time. In this paper, we propose a convex reformulation less general than MIQCR because it is limited to the general integer case, but that has a significantly smaller size. We call this approach Compact Quadratic Convex Reformulation (CQCR). We evaluate CQCR from the computational point of view. We perform our experiments on instances of general integer quadratic programs with one equality constraint. We show that CQCR is much faster than MIQCR and than the general non-linear solver BARON (Sahinidis and Tawarmalani, User's manual, 2010) to solve these instances. Then, we consider the particular class of binary quadratic programs. We compare MIQCR and CQCR on instances of the Constrained Task Assignment Problem. These experiments show that CQCR can solve instances that MIQCR and other existing methods fail to solve.

Currently used methods for the analysis of contingency tables with log-linear models and their extensions are essentially a way to subject the parameters of the saturated model to a set of linear constraints. However, in many contexts, expecially when a subset of the variables are ordinal, it would be very useful to be able to consider also linear inequality constraints. In this paper we show how to extend previous models for contingency tables by considering simultaneously both equality constraints. We also give a self contained discussion of the problem of testing equality and inequality constraints in the context of contingency tables analysis.

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.

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.

In this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints, in which every problem of the class is obtained by linear perturbations of the objective function, while the “core” objective function and the constraint functions are kept fixed. Assume that the linear independence constraint qualification is satisfied at every point in the constraint set. It is shown that almost all problems in the class are such that (i) the restriction of the objective function on the constraint set is coercive and regular at infinity; (ii) there is a unique optimal solution, lying on a unique active manifold, at which the strict complementarity and second order sufficiency conditions, the quadratic growth condition, and the Hölder type global error bound hold, and (iii) all minimizing sequences converge. Furthermore, the active manifold is constant, and the optimal solution and the optimal value function depend analytically under local perturbations of the objective function. These results are combined with a standard result about the existence of sums of squares certificates to prove that we can build a sequence of semidefinite programs whose solutions give rise to a sequence of points converging to the optimal solution of the original problem in finitely many steps. It is worth emphasizing that the results of this study hold globally and we do not require the problem to be convex or the constraint set to be compact.

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.

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set. The objective function is defined by a sum of local objective functions, while the global constraint set is produced by the intersection of local constraint sets. In particular, we study two cases: one where the equality constraint is absent, and the other where the local constraint sets are identical. We devise two distributed primal-dual subgradient algorithms based on the characterization of the primal-dual optimal solutions as the saddle points of the Lagrangian and penalty functions. These algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the Slater's condition.

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.