Dantzig–Wolfe reformulations for binary quadratic problems

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.

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.

A novel and effective method for the design of maximum directivity irregular phased arrays is proposed in this article. By decomposing the radiation pattern of irregular arrays (IAs) into the product of subarray pattern and subarray array factor, the principal factor for the reduction of directivity and the appearance of high sidelobes is analyzed. Aimed at maximizing directivity of IAs, a binary quadratic optimization problem (BQOP) is established for the design of subarray tiling configuration, where the directivities at considered scanning angles satisfying given peak sidelobes level (SLL) is maximized. To consider the directivities at all the scanning angles within given scan ranges, a concept of “generalized directivity” is introduced and defined. Meanwhile, it is known that high directivity usually enables to suppress high SLL to a great extent. Therefore, the BQOP is reduced to a simplified binary quadratic problem (SBQP) for the maximization of the “generalized directivity” without the high-dimensional constraint on SLL. The associated complex excitations at subarray level are solved using convex optimization after the subarray tiling configuration is determined. By comparing to the reported state-of-the-art methods, several representative numerical examples are implemented to assess the effectiveness of the proposed method.

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.

A network approach for specially structured linear programs arising in 0–1 quadratic optimization

The dataset contains instances for the cardinality constrained quadratic knapsack problem. These are used to test decomposition methods for Binary Quadratic Programs. Full details are given in the corresponding paper Dantzig-Wolfe Reformulations for Binary Quadratic Problems . The dataset contains three sets of instances - qcr_instances: base instance, together with the optimal quadratic convex reformulation (QCR) multipliers found by solving an associated semidefinite program - mqcr_instances: base instance, together with the optimal improved convex 0-1 quadratic program reformulation (MQCR) multipliers found by solving an associated semidefinite program - convexity_analysis: instances in which the objective function quadratic cost matrix has a given number of positive eigenvalues.

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.

A column generation approach for the unconstrained binary quadratic programming problem

The unconstrained binary quadratic minimization problem is known to be NP-hard and due to its computational challenge and application capability, it becomes more and more considered and involved by the recent research studies, including both exact and heuristic solution approaches. Our work is in line with what was suggested by Glover et al. (in Eur. J. Oper. Res. 137, 272–287, 2002) who proposed one pass heuristics as alternatives to the well-known Devour Digest Tidy-up procedure (DDT) of Boros et al. (in RRR 39-89, 1989). The “devour” step sets a term of the current representation to 0 or 1, and the “tidy-up” step substitutes the logical consequences derived from the “digest” step into the current quadratic function. We propose several versions of the DDT constructive heuristic based on the alternative representation of the quadratic function. We also present an efficient implementation of local search using one-flip and two-flip moves that simultaneously change the values of one or two binary variables. Computational experiments performed on instances up to 7000 variables show the efficiency of our implementation in terms of quality improvement and CPU use enhancement.

A linearization method for mixed 0–1 polynomial programs

This article presents an exploratory landscape analysis of three NP-hard combinatorial optimisation problems: the number partitioning problem, the binary knapsack problem, and the quadratic binary knapsack problem. In the article, we examine empirically a number of fitness landscape properties of randomly generated instances of these problems. We believe that the studied properties give insight into the structure of the problem landscape and can be representative of the problem difficulty, in particular with respect to local search algorithms. Our work focuses on studying how these properties vary with different values of problem parameters. We also compare these properties across various landscapes that were induced by different penalty functions and different neighbourhood operators. Unlike existing studies of these problems, we study instances generated at random from various distributions. We found a general trend where some of the landscape features in all of the three problems were found to vary between the different distributions. We captured this variation by a single, easy to calculate parameter and we showed that it has a potentially useful application in guiding the choice of the neighbourhood operator of some local search heuristics.

A simplified binary artificial fish swarm algorithm for 0–1 quadratic knapsack problems

Various conic relaxations of quadratic optimization problems in nonnegative variables for combinatorial optimization problems, such as the binary integer quadratic problem, quadratic assignment problem (QAP), and maximum stable set problem have been proposed over the years. The binary and complementarity conditions of the combinatorial optimization problems can be expressed in several ways, each of which results in different conic relaxations. For the completely positive, doubly nonnegative and semidefinite relaxations of the combinatorial optimization problems, we discuss the equivalences and differences among the relaxations by investigating the feasible regions obtained from different representations of the combinatorial condition which we propose as a generalization of the binary and complementarity condition. We also study theoretically the issue of the primal and dual nondegeneracy, the existence of an interior solution and the size of the relaxations, as a result of different representations of the combinatorial condition. These characteristics of the conic relaxations affect the numerical efficiency and stability of the solver used to solve them. We illustrate the theoretical results with numerical experiments on QAP instances solved by SDPT3, SDPNAL+ and the bisection and projection method.

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

The Quadratic Knapsack Problem (QKP) deals with maximizing a quadratic objective function subject to given constraints on the capacity of the Knapsack. We assume all coefficients to be non-negative and all variables to be binary. Solution to QKP generalizes the problem of finding whether a graph contains a clique of given size. We propose in this paper a Novel Quantum Evolutionary Algorithm (NQEA) for QKPs. These algorithms are general enough and can be used for similar subsection of problems. We report in this paper solutions which lie in less than 1% of the optimal solutions. We also show that our algorithm is scalable to much larger problem sizes and is capable of exploiting the search space to its maximum.