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.
The purpose of this paper is to provide strong reformulations for binary quadratic problems. We propose a first methodological analysis on a family of reformulations combining Dantzig–Wolfe and ...Quadratic Convex optimization principles. We show that a few reformulations of our family yield continuous relaxations that are strong in terms of dual bounds and computationally efficient to optimize. As a representative case study, we apply them to a cardinality constrained quadratic knapsack problem, providing extensive experimental insights. We report and analyze in depth a particular reformulation providing continuous relaxations whose solutions turn out to be integer optima in all our tests.
•We investigate two equity-concerned dispersion problems: max-minsum and min-diffsum.•We present ILP formulations and we discuss their strengthening.•We propose construction heuristics and a local ...search metaheuristic.•We investigate the key features of these algorithms.•We test our algorithms on publicly available benchmark instances up to 500 elements.
Given a set N, a pairwise distance function d and an integer number m, the Dispersion Problems (DPs) require to extract from N a subset M of cardinality m, so as to optimize a suitable function of the distances between the elements in M. Different functions give rise to a whole family of combinatorial optimization problems. In particular, the max-sum DP and the max-min DP have received strong attention in the literature. Other problems (e.g., the max-minsum DP and the min-diffsum DP) have been recently proposed with the aim to model the optimization of equity requirements, as opposed to that of more classical efficiency requirements. Building on the main ideas which underly some state-of-the-art methods for the max-sum DP and the max-min DP, this work proposes some constructive procedures and a Tabu Search algorithm for the new problems. In particular, we investigate the extension to the new context of key features such as initialization, tenure management and diversification mechanisms. The computational experiments show that the algorithms applying these ideas perform effectively on the publicly available benchmarks, but also that there are some interesting differences with respect to the DPs more studied in the literature. As a result of this investigation, we also provide optimal results and bounds as a useful reference for further studies.
Binary quadratic programming (BQP) is a typical integer programming problem widely applied in the field of signal processing, economy, management and engineering. However, it is NP-hard and lacks ...efficient algorithms. Due to this reason, in this paper, some novel polynomial algorithms are proposed to solve a class of unconstrained and linearly constrained binary quadratic programming problems. We first deduce the polynomial time solvable algorithms to the unconstrained binary quadratic programming problems with Q being a seven-diagonal matrix (UBQP7) and a five-diagonal matrix (UBQP5) respectively with two different approaches. Then, the algorithm to unconstrained problem is combined with the dynamic programming method to solve the linearly constrained binary quadratic programming problem with Q being a five-diagonal matrix (LCBQP5). In addition, the polynomial solvable feature of these algorithms is analyzed and some specific examples are presented to illustrate these new algorithms. Lastly, we demonstrate their polynomial feature as well as their high efficiency.
Given an undirected graph with costs associated to its edges and pairs of edges, the Quadratic Minimum Spanning Tree Problem (QMSTP) requires to determine a spanning tree of minimum total cost. This ...is a proper model for network problems in which both routing and interference costs need to be considered. It is NP-hard in the strong sense and not approximable unless P=NP. This paper describes a Tabu Search algorithm, with two independent and adaptively tuned tabu lists, and a Variable Neighbourhood Search algorithm. Both metaheuristics are based on the same neighbourhood, but the Tabu Search proves more effective and robust than the Variable Neighbourhood Search. To assess the quality of these results, we provide a comparison with the heuristic algorithms proposed in the recent literature and we reimplement, with minor improvements, an exact algorithm drawn from the literature, which confirms the optimality of the results obtained on small instances.
This book is a printed edition of the Special Issue Advances in Intelligent Vehicle Control that was published in the journal Sensors. It presents a collection of eleven papers that covers a range of ...topics, such as the development of intelligent control algorithms for active safety systems, smart sensors, and intelligent and efficient driving. The contributions presented in these papers can serve as useful tools for researchers who are interested in new vehicle technology and in the improvement of vehicle control systems.
Glover’s linearization technique is revisited for solving the binary quadratic programming problem with a budget constraint (BBQP). When compared with the recent two linearizations for (BBQP), it not ...only provides a tighter relaxation at the root node, but also has a much better computational performance for globally solving (BBQP).
The maximum diversity problem (MDP) is a challenging NP-hard problem with a wide range of real applications. Several researchers have pointed out close relationship between the MDP and unconstrained ...binary quadratic program (UBQP). In this paper, we provide procedures to solve MDP ideas from the UBQP formulation of the problem. We first give some local optimality results for r-flip improvement procedures on MDP. Then, a set of highly effective diversification approaches based on sequential improvement steps for MDP are presented. Four versions of the approaches are used within a simple tabu search and applied to 140 benchmark MDP problems available on the Internet. The procedures solve all 80 small- to medium-sized problems instantly to the best known solutions. For 22 of the 60 large problems, the procedures improved by significant amounts the best known solutions in reasonably short CPU time.
We introduce and prove new necessary and sufficient conditions to carry out a compact linearization approach for a general class of binary quadratic problems subject to assignment constraints that ...has been proposed by Liberti (4OR 5(3):231–245,
2007
,
https://doi.org/10.1007/s10288-006-0015-3
). The new conditions resolve inconsistencies that can occur when the original method is used. We also present a mixed-integer linear program to compute a minimally sized linearization. When all the assignment constraints have non-overlapping variable support, this program is shown to have a totally unimodular constraint matrix. Finally, we give a polynomial-time combinatorial algorithm that is exact in this case and can be used as a heuristic otherwise.
In recent years, the general binary quadratic programming (BQP) model has been widely applied to solve a number of combinatorial optimization problems. In this paper, we recast the maximum vertex ...weight clique problem (MVWCP) into this model which is then solved by a probabilistic tabu search algorithm designed for the BQP. Experimental results on 80 challenging DIMACS-W and 40 BHOSLIB-W benchmark instances demonstrate that this general approach is viable for solving the MVWCP problem.
PDF
