Algorithm Portfolios for Solving the Quadratic Assignment Problem

-Quadratic assignment problems (QAPs) are NP-hard combinatorial optimization problems often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator whose instances can be used as benchmark for meta-heuristics. Our generator aggregates various QAP instances into a larger size QAP instance in order to obtain an large size, additively decomposable QAP. We assume that a QAP instance that is difficult for local search has many local optima from which local search needs to escape from. We call the resulting QAPs the composite QAP instances (cQAPs). We use numerical and empirical techniques for the landscape analysis of generated composite QAPs. The comparison between our QAP instances with the other QAPs from the literature classify cQAPs as difficult. We show that heuristic algorithms that exploit the particularities of the cQAP search space, like iterated local search, can outperform other heuristics that do not consider the structure of this search space, like the multi-restart local search.

In this paper, we describe an implementation of the iterated tabu search (ITS) algorithm for the quadratic assignment problem (QAP), which is one of the well-known problems in combinatorial optimization. The medium- and large-scale QAPs are not, to this date, practically solvable to optimality, therefore heuristic algorithms are widely used. In the proposed ITS approach, intensification and diversification mechanisms are combined in a proper way. The goal of intensification is to search for good solutions in the neighbourhood of a given solution, while diversification is responsible for escaping from local optima and moving towards new regions of the search space. In particular, the following enhancements were implemented: new formula for fast evaluation of the objective function and efficient data structure; extended intensification mechanisms (including randomized tabu criterion, combination of tabu search and local search, dynamic tabu list maintaining); enhanced diversification strategy using periodic tabu tenure and special mutation procedure. The ITS algorithm is tested on the different instances taken from the QAP library QAPLIB. The results from the experiments demonstrate promising efficiency of the proposed algorithm, especially for the random QAP instances.

Quadratic assignment problems (QAPs) is a NP-hard combinatorial optimization problem. QAPs are often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator that can be used for benchmarking for heuristic algorithms. Our QAP generator combines small size QAPs with known optimum solution into a larger size QAP instance. We call these instances composite QAPs (cQAPs), and we show that the cost function of cQAPs is additively decomposable. We give mild conditions for which a cQAP instance has known optimum solution. We generate cQAP instances using uniform distributions with different bounds for the component QAPs and for the rest of the cQAP elements. Numerical and analytical techniques that measure the difficulty of the cQAP instances in comparison with other QAPs from the literature are introduced. These methods point out that some cQAP instances are difficult for local search with many local optimum of various values, low epistasis and non-trivial asymptotic behaviour.

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.

A new tool for automated transformation of Quadratic Assignment Problem instances to Quadratic Unconstrained Binary Optimisation models

Differential evolution (DE) was introduced by Stone and Price in 1995 as a population-based stochastic search technique for solving optimization problems in a continuous space. DE has been successfully applied to various real world numerical optimization problems. In recent years not only continuous real-valued function, the applications of DE on combinatorial optimization problems with discrete decision variables are reported. However, genetic operator in the standard DE can not directly applied to discrete space. In this paper, we propose a method to solve quadratic assignment problems (QAP) by DE. The QAP is a well-known combinatorial optimization problem with a wide variety of practical applications. It is NP-hard and is considered to be one of the most difficult problems. In the QAP, a candidate solution can represented a permutation of integer. The proposed method employs permutation representation for individuals in DE. Therefore, a individual vector is encoded directly as a permutation. In discrete space, to realize efficient solution search like standard DE which have continuous nature, we modify differential operator to handle permutation encoding. Additionally, in order to maintain diversity of population, restart strategy and tabu list are introduced to proposed method instead of crossover operator. Finally, we show the experimental results using instances of QAPLIB and the efficacy of proposed method.

The facility layout design problem is an extensively studied research problem and belongs to nonpolynomial hard (NP-hard) combinatorial optimization problem. Quadratic assignment problem (QAP) is one of the formulations that is investigated for facility layout design because of its wide applicability. Ant colony optimization (ACO), a biologically inspired heuristic has centered on solving the QAP by achieving approximation as good as possible. This paper presents a population-based hybrid ant system (PHAS), which is an extension of the hybrid ant system (HAS) in which the size of the ant colony has been fixed. The performance of the proposed ant algorithm for QAP is compared with the existing metaheuristic implementations such as tabu search, reactive tabu search, simulated annealing, genetic hybrid method, HAS, and max–min ant system. The experimental results show that the proposed PHAS perform significantly better than the other existing algorithms of QAP.

The quadratic assignment problem (QAP) is one of the NP-hard combinatorial optimization problems, which is very difficult to solve. The tabu search technique, one of the various heuristic methods for the QAP, has been implemented in a neural network form with chaotic dynamics. In order to achieve a high-speed massively parallel solution of the QAP, a mixed analog/digital integrated circuit implementation of the system is mandatory. However, the algorithm in Hasegawa et al. (1997) cannot be implemented with IC technology as it is. Therefore, the algorithm is modified to be suitable for the analog IC implementation in this paper. The performance of the modified algorithm is investigated with numerical simulations. The circuit characteristics obtained from a prototype chip is extensively used in the simulations.

The quadratic assignment problem (QAP) is a well‐known challenging combinatorial optimization problem that has received many researchers' attention with varied real‐world and industrial applications areas. It is noteworthy to mention that a plethora of nature‐inspired optimization algorithms have successfully been used to solve various optimization problems, including several variants of the QAPs. In this article, a comprehensive literature review is presented to show the most relevant nature‐inspired algorithms that have been used in solving the QAP. More so, extensive experiments are conducted and analyzed to show the performance of the well‐known state‐of‐the‐art nature‐inspired meta‐heuristic optimization algorithms in solving the QAP, including the ant colony optimization (ACO), bat algorithm, genetic algorithm (GA), particle swarm optimization (PSO), and tabu search algorithm. Besides, a modified variant of the discrete PSO algorithm is implemented and compared with existing approaches. The six selected algorithms' performances, including the modified PSO, are validated on eight commonly used QAP instances of varying complexity and size, considering the quality of solutions achieved and computational time consumed by the representative algorithms. The numerical results revealed that the most competitive algorithm was ACO, while the GA appeared to be the worst performed algorithm among the six compared meta‐heuristic algorithms. However, based on the extensive analysis conducted on the tested algorithms, further improvements are suggested, including implementing new modified versions of the tested algorithms to tackle the QAP and its variant instances.

Parameter Selection of Discrete Particle Swarm Optimization Algorithm for the Quadratic Assignment Problems

Recently, an interesting analogy between problems in combinatorial optimization and statistical mechanics has been developed and has proven useful in solving certain traditional optimization problems such as computer design, partitioning, component placement, wiring, and traveling salesman problems. The analogy has resulted in a methodology, termed "simulated annealing," which, in the process of iterating to an optimum, uses Monte Carlo sampling to occasionally accept solutions to discrete optimization problems which increase rather than decrease the objective function value. This process is counter to the normal 'steepest-descent' algorithmic approach. However, it is argued in the analogy that by taking such controlled uphill steps, the optimizing algorithm need not get "stuck" on inferior solutions. This paper presents an application of the simulated annealing method to solve the quadratic assignment problem (QAP). Performance is tested on a set of "standard" problems, as well as some newly generated larger problems (n = 50 and n = 100). The results are compared to those from other traditional heuristics, e.g., CRAFT, biased sampling, and a revised Hillier procedure. It is shown that under certain conditions simulated annealing can yield higher quality (lower cost) solutions at comparable CPU times. However, the simulated annealing algorithm is sensitive to a number of parameters, some of whose effects are investigated and reported herein through the analysis of an experimental design.

The power of genetic algorithms (GAs) has been demonstrated for various domains of the computer science, including combinatorial optimization. In this paper, we propose a new conceptual modification of the genetic algorithm entitled a “restart-based genetic algorithm” (RGA). An effective implementation of RGA for a well-known combinatorial optimization problem, the quadratic assignment problem (QAP), is discussed. The results obtained from the computational experiments on the QAP instances from the publicly available library QAPLIB show excellent performance of RGA. This is especially true for the real-life like QAPs.

Graph matching problem that incorporates pair-wise constraints can be formulated as Quadratic Assignment Problem(QAP). The optimal solution of QAP is discrete and combinational, which makes QAP problem NP-hard. Thus, many algorithms have been proposed to find approximate solutions. In this paper, we propose a new algorithm, called Nonnegative Orthogonal Graph Matching (NOGM), for QAP matching problem. NOGM is motivated by our new observation that the discrete mapping constraint of QAP can be equivalently encoded by a nonnegative orthogonal constraint which is much easier to implement computationally. Based on this observation, we develop an effective multiplicative update algorithm to solve NOGM and thus can find an effective approximate solution for QAP problem. Comparing with many traditional continuous methods which usually obtain continuous solutions and should be further discretized, NOGM can obtain a sparse solution and thus incorporates the desirable discrete constraint naturally in its optimization. Promising experimental results demonstrate benefits of NOGM algorithm.

The quadratic assignment problem (QAP) is an NP-hard problem with a wide range of applications in many real-world applications. This study introduces a discrete rat swarm optimizer (DRSO)algorithm for the first time as a solution to the QAP and demonstrates its effectiveness in terms of solution quality and computational efficiency. To address the combinatorial nature of the QAP, a mapping strategy is introduced to convert real values into discrete values, and mathematical operators are redefined to make then suitable for combinatorial problems. Additionally, a solution quality improvement strategy based on local search heuristics such as 2-opt and 3-opt is proposed. Simulations with test instances from the QAPLIB test library validate the effectiveness of the DRSO algorithm, and statistical analysis using the Wilcoxon parametric test confirms its performance. Comparative analysis with other algorithms demonstrates the superior performance of DRSO in terms of solution quality, convergence speed, and deviation from the best-known values, making it a promising approach for solving the QAP.

Applying the attribute based hill climber heuristic to the vehicle routing problem