Binary Quadratic Problem decomposition methods - kQKP dataset

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.

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.

A column generation approach for the unconstrained binary quadratic programming problem

Many combinatorial optimization problems can be formulated as the minimization of a 0-1 quadratic function subject to linear constraints. In this paper, we are interested in the exact solution of this problem through a two-phase general scheme. The first phase consists in reformulating the initial problem either into a compact mixed integer linear program or into a 0-1 quadratic convex program. The second phase simply consists in submitting the reformulated problem to a standard solver. The efficiency of this scheme strongly depends on the quality of the reformulation obtained in phase 1. We show that a good compact linear reformulation can be obtained by solving a continuous linear relaxation of the initial problem. We also show that a good quadratic convex reformulation can be obtained by solving a semidefinite relaxation. In both cases, the obtained reformulation profits from the quality of the underlying relaxation. Hence, the proposed scheme gets around, in a sense, the difficulty to incorporate these costly relaxations in a branch-and-bound algorithm.

Invariants of SDP exactness in quadratic programming

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.

The quadratic shortest path problem (QSPP) has a lot of applications in real-life problems. This thesis consists of four essays about the QSPP concerning its theory and computation, where linear and semidefinite programming were used as a major tool for solving and understanding the problem. In the second chapter, we investigate the computational complexity of the problem as well as the polynomial-time solvable special cases. In the third and fourth chapters, we show that the linearization problem of the quadratic shortest path problem on directed acyclic graphs can be solved efficiently. By using the linearization problem of binary quadratic problems as a unifying tool, the dominance between the Generalized Gilmore-Lawler bound and the first level RLT bound is also established. We also propose a new class of relaxation for binary quadratic optimization problems based on its linearization problem. In the last chapter, we derive several semidefinite programming relaxations with increasing complexity for the quadratic shortest path problem. We implement the alternating direction method of multipliers to solve our strongest semidefinite programming relaxation, and the obtained bound is currently the best one solved in the literature.

The aim of this paper is to present a review on the state-of-the-art related to quadratic programming (QP) according to the methodology of Kitchenham and Charter. Our review was motivated from four questions: Q1: In what areas are quadratic programming models used? Q2: What optimization methods are used to solve quadratic programming problems? Q3: What optimization methods or algorithms have been used for portfolio selection? Q4: What optimization methods or algorithms solve quasiconvex quadratic problems? We obtain very interesting results on QP and discovered a new line of research for future investigations.KeywordsQuadratic programmingOptimization methodsQuasiconvex quadratic problemsPortfolio selection problems

Because it is classified as NP-hard, the binary knapsack problem is a good example of a combinatorial optimization problem that still presents increased difficulty when attempting to determine the optimal solution for any instance. Although exact and heuristic methods have been developed in an attempt to solve the problem, such methods have been unable to solve even small instances of the problem. In this paper, new algorithms for this problem are automatically generated by means of genetic programming from sets of training instances of different sizes and are then evaluated against other larger sized sets of instances, thereby detecting the robustness of the algorithms for larger instances. Overall, the produced algorithms are able to identify up to 52% of the optimal solutions for the biggest instances used.

In this paper, we propose a new multiuser detector that is based on the explicit estimation of Lagrange multipliers. The optimum multiuser detector, proposed by S. Verdu in 1986, is a binary quadratic problem. It belongs to NP hard problems and can not be used currently. In this paper, the binary quadratic problem is changed into a quadratic optimization problem with equation constraints, and then written in Lagrange form, the relationship between the Lagrange multipliers, user information codes and channel noise is derived. According to different methods of estimating Lagrange multipliers, two existing multiuser detectors are given: decorrelation and minimum mean-squared error (MMSE) detectors. According to the special features of the multiuser detection problem, a new method of estimating Lagrange multipliers is given in this paper and then a new multiuser detector is obtained. The numerical results showed that the bit-error-ratio (BER) of the new detector is close to that of the optimum multiuser detector and less than that of sub-optimum multiuser detection.

The problem of finding the best rank-one approximation to higher-order tensors has extensive engineering and statistical applications. It is well-known that this problem is equivalent to a homogeneous polynomial optimization problem. In this paper, we study theoretical results and numerical methods of this problem, particularly focusing on the 4-th order symmetric tensor case. First, we reformulate the polynomial optimization problem to a matrix programming, and show the equivalence between these two problems. Then, we prove that there is no duality gap between the reformulation and its Lagrangian dual problem. Concerning the approaches to deal with the problem, we propose two relaxed models. The first one is a convex quadratic matrix optimization problem regularized by the nuclear norm, while the second one is a quadratic matrix programming regularized by a truncated nuclear norm, which is a D.C. function and therefore is nonconvex. To overcome the difficulty of solving this nonconvex problem, we approximate the nonconvex penalty by a convex term. We propose to use the proximal augmented Lagrangian method to solve these two relaxed models. In order to obtain a global solution, we propose an alternating least eigenvalue method after solving the relaxed models and prove its convergence. Numerical results presented in the last demonstrate, especially for nonpositive tensors, the effectiveness and efficiency of our proposed methods.

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 consider a chance-constrained quadratic knapsack problem (CQKP) where each item has a random size that is finitely distributed. We present a new convex 0-1 quadratic program reformulation for CQKP. This new reformulation improves the existing reformulation for general 0-1 quadratic program based on diagonal perturbation in the sense that the continuous relaxation of the new reformulation is tighter than or at least as tight as that of the existing reformulation. The improved reformulation is derived from a general matrix decomposition of the objective function and piecewise linearization of 0-1 variables. We show that the optimal parameters in the improved reformulation can be obtained by solving an SDP problem. Extension to k-item probabilistic quadratic knapsack problems is also discussed. Preliminary comparison results are reported to demonstrate the effectiveness of the improved reformulation.

In this paper, we propose an reactive method to solve the Quadratic Knapsack Problem (noted QKP). The quadratic knapsack problem is a well-studied combinatorial optimisation problem. In all variants of the quadratic knapsack problems, for a set of given items, profits are not only assigned to individual items but also to pairs of them. The pairwise profit is added to the quadratic objective value only when the two corresponding items are both included in the same knapsack. We let a knapsack with a capacity C and a set of candidate objects (or items), each item has a positive weight w <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , and profit Pi, if selected, generates an object profit p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and a pairwise profit P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</inf> with any other selected object j. The objective of the QKP is to select a subset of objects to fill the knapsack so as to maximize the overall profit P, while the overall weight does not exceed the knapsack capacity C. This problem is known as quadratic knapsack problem and has been shown strongly NP-hard and it has been applied with a variety of important applications, such as, in the location of satellites, airports, railway stations or freight terminals. The proposed method is mainly based on two complementary phases. In the first phase we use a greedy algorithm for provide the starting solution. In the second phase we improved the quality of the starting solutions based on a reactive search with applying a destroy and repair process, the reactive phase is based both diversification and intensification strategy. The obtained results are compared to those reached by the Cplex solverl and literature. Consequently, the experimental results show the effectiveness of the proposed approach, and the computational results show that the algorithm is capable of solving instances of the QKP that cannot be solved by other methods.

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.