Solving Quadratic Assignment Problems by ‘Simulated Annealing’

Solving the traveling salesman problem based on an adaptive simulated annealing algorithm with greedy search

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

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.

The probabilistic travelling salesman problem (PTSP) is an extension of the well-known travelling salesman problem (TSP), which has been extensively studied in the field of combinatorial optimization. The goal of the TSP is to find the minimum length of a tour to all customers, given the distances between all pairs of customers whereas the objective of the PTSP is to minimize the expected length of the a priori tour where each customer requires a visit only with a given probability (Bertsimas, 1988; Bertsimas et al., 1990; Jaillet, 1985). The main difference between the PTSP and the TSP is that in the PTSP the probability of each node being visited is between 0.0 and 1.0 while in TSP the probability of each node being visited is 1.0. Due to the fact that the element of uncertainty not only exists, but also significantly affects the system performance in many real-world transportation and logistics applications, the results from the PTSP can provide insights into research in other probabilistic combinatorial optimization problems. Moreover, the PTSP can also be used to model many real-world applications in logistical and transportation planning, such as daily pickup-delivery services with stochastic demand, job sequencing involving changeover cost, design of retrieval sequences in a warehouse or in a cargo terminal operations, meals on wheels in senior citizen services, trip-chaining activities, vehicle routing problem with stochastic demand, and home delivery service under e-commerce (Bartholdi et al., 1983; Bertsimas et al., 1995; Campbell, 2006; Jaillet, 1988; Tang & Miller-Hooks, 2004). Early PTSP computational studies, dating from 1985, adopted heuristic approaches that were modified from the TSP (e.g., nearest neighbor, savings approach, spacefilling curve, radial sorting, 1-shift, and 2-opt exchanges) (Bartholdi & Platzman, 1988; Bertsimas, 1988; Bertsimas & Howell, 1993; Jaillet, 1985, 1987; Rossi & Gavioli, 1987). With its less than satisfactory performance in yielding solution quality, researchers in the recent years switch to metaheuristic methods, such as ant colony optimization (Bianchi, 2006; Branke & Guntsch, 2004), evolutionary algorithm (Liu et al., 2007), simulated annealing (Bowler et al., 2003), threshold accepting (Tang & Miller-Hooks, 2004) and scatter search (Liu, 2006, 2007, 2008). Because the genetic algorithm (GA), a conceptual framework of the population-based metaheuristic method, has been shown to yield promising outcomes for solving various complicated optimization problems in the past three decades (Back et al., 1997; Davis, 1991; O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

Revisiting simulated annealing: A component-based analysis

Introduction. The quadratic assignment problem is a well-established NP-hard problem in combinatorial optimization with applications in diverse fields like economics, archaeology, and chemistry. Due to its complexity, research on efficient solution methods remains active, including efforts for parallelization on multiprocessor computing systems. However, effective parallel algorithms are crucial to fully leverage these computational resources. In this context, algorithm unions (portfolios and teams) play a significant role in achieving parallel execution for solving such problems. Research objectives. This work investigates the application of portfolios constructed from modifications of the repeated iterated tabu search algorithm to the quadratic assignment problem. The effectiveness of these portfolios was evaluated through experimental computations. Results. The portfolios, derived from modifications of the repeated iterated tabu search algorithm, were applied to the quadratic assignment problem. For the most demanding test instances, the proposed algorithms were evaluated on the SCIT-4 supercomputer, alongside previously published results from the authors, confirming their competitive performance. Additionally, we assessed the parallel efficiency of these portfolios in solving instances of the quadratic assignment problem. The results demonstrate their ability to accelerate the optimization process (with speedup dependent on problem size and utilized processors), enabling the solution of large-scale problems. Conclusions. The conducted studies demonstrate that employing algorithm portfolios significantly accelerates the solution process for the quadratic assignment problem. Analysis of the results reveals a near-linear speedup factor achieved by the portfolio. For the challenging test instance tai100a, a new best solution value of 21040996 was obtained using a portfolio of 16 algorithms. Keywords: quadratic assignment problem, algorithm portfolios, experimental research, algorithm portfolios efficiency.

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 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.

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

The traveling salesman problem (TSP) is one of the most studied problems in combinatorial optimization. Given a set of nodes and the distances between them, it consists in finding the shortest route that visits each node exactly once and returns to the first. Nevertheless, more flexible and applicable formulations of this problem exist and can be considered. The Steiner TSP (STSP) is a variant of the TSP that assumes that only a given subset of nodes must be visited by the shortest route, eventually visiting some nodes and edges more than once. In this paper, we adapt some classical TSP constructive heuristics and neighborhood structures to the STSP variant. In particular, we propose a reduced 2‐opt neighborhood and we show that it leads to better results in smaller computation times. Computational results with an implementation of a GRASP heuristic using path‐relinking and restarts are reported. In addition, ten large test instances are generated. All instances and their best‐known solutions are made available for download and benchmarking purposes.

The Traveling Salesman Problem (TSP) is often cited as the prototypical “hard” combinatorial optimization problem. As such, it would seem to be an ideal candidate for nonstandard algorithmic approaches, such as simulated annealing, and, more recently, genetic algorithms. Both of these approaches can be viewed as variants on the traditional technique called local optimization. This paper surveys the state of the art with respect to the TSP, with emphasis on the performance of traditional local optimization algorithms and their new competitors, and on what insights complexity theory does, or does not, provide.

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.

Traveling salesman problem with drone under recharging policy

The traveling salesman problem (TSP) belongs to the most basic, most important, and most investigated problems in combinatorial optimization. Although it is an ${\cal NP}$-hard problem, many of its special cases can be solved efficiently in polynomial time. We survey these special cases with emphasis on the results that have been obtained during the decade 1985--1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler, and Shmoys [The Traveling Salesman Problem---A Guided Tour of Combinatorial Optimization, Wiley, Chichester, pp. 87--143].

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems, widely studied due to its numerous industrial applications in logistics, supply chain management, and resource planning. Over the years, various metaheuristic algorithms have been developed to provide near-optimal solutions within a reasonable computational time. While traditional methods such as Genetic Algorithms (GA) and Simulated Annealing (SA) have been applied widely to the TSP, recent developments in metaheuristics have introduced new methods that can ensure improved performance. In this study, we compare three new metaheuristics: Arctic Puffin Optimization (APO), Human Evolutionary Optimization (HEO), and Ship Rescue Optimization (SRO). These methods, inspired by puffin's behavior, human evolutions, and naval rescue, respectively, have demonstrated high optimization capability in various areas. However, these have not been extensively tried on solving the TSP yet. This research seeks to establish the performance of these algorithms by comparing their convergence rate, solution quality, and computational efficiency with well-established metaheuristics. The results of the experiment expose the strengths and weaknesses of such innovative approaches, showing their potential as competitive alternatives to solve complex combinatorial optimization problems