Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

20 years of Greedy Randomized Adaptive Search Procedures with Path Relinking

This is a comprehensive review of the Greedy Randomized Adaptive Search Procedure (GRASP) metaheuristic and its hybridization with Path Relinking (PR) over the past two decades. GRASP with PR has become a widely adopted approach for solving hard optimization problems since its proposal in 1999. The paper covers the historical development of GRASP with PR and its theoretical foundations, as well as recent advances in its implementation and application. The review includes a critical analysis of variants of PR, including memory-based and randomized designs, with a total of ten different implementations. It describes these advanced designs both theoretically and practically on two well-known optimization problems, linear ordering and max-cut. The paper also explores the hybridization of GRASP with PR and other metaheuristics, such as Tabu Search and Scatter Search. Overall, this review provides valuable insights for researchers and practitioners seeking to utilize GRASP with PR for solving optimization problems.Comment: 28 pages, 13 figure

A multi-start biased-randomized algorithm for the capacitated dispersion problem

The capacitated dispersion problem is a variant of the maximum diversity problem in which a set of elements in a network must be determined. These elements might represent, for instance, facilities in a logistics network or transmission devices in a telecommunication network. Usually, it is considered that each element is limited in its servicing capacity. Hence, given a set of possible locations, the capacitated dispersion problem consists of selecting a subset that maximizes the minimum distance between any pair of elements while reaching an aggregated servicing capacity. Since this servicing capacity is a highly usual constraint in real-world problems, the capacitated dispersion problem is often a more realistic approach than is the traditional maximum diversity problem. Given that the capacitated dispersion problem is an NP-hard problem, whenever large-sized instances are considered, we need to use heuristic-based algorithms to obtain high-quality solutions in reasonable computational times. Accordingly, this work proposes a multi-start biased-randomized algorithm to efficiently solve the capacitated dispersion problem. A series of computational experiments is conducted employing small-, medium-, and large-sized instances. Our results are compared with the best-known solutions reported in the literature, some of which have been proven to be optimal. Our proposed approach is proven to be highly competitive, as it achieves either optimal or near-optimal solutions and outperforms the non-optimal best-known solutions in many cases. Finally, a sensitive analysis considering different levels of the minimum aggregate capacity is performed as well to complete our study.Peer ReviewedPostprint (published version

Incorporating Memory and Learning Mechanisms Into Meta-RaPS

Due to the rapid increase of dimensions and complexity of real life problems, it has become more difficult to find optimal solutions using only exact mathematical methods. The need to find near-optimal solutions in an acceptable amount of time is a challenge when developing more sophisticated approaches. A proper answer to this challenge can be through the implementation of metaheuristic approaches. However, a more powerful answer might be reached by incorporating intelligence into metaheuristics. Meta-RaPS (Metaheuristic for Randomized Priority Search) is a metaheuristic that creates high quality solutions for discrete optimization problems. It is proposed that incorporating memory and learning mechanisms into Meta-RaPS, which is currently classified as a memoryless metaheuristic, can help the algorithm produce higher quality results. The proposed Meta-RaPS versions were created by taking different perspectives of learning. The first approach taken is Estimation of Distribution Algorithms (EDA), a stochastic learning technique that creates a probability distribution for each decision variable to generate new solutions. The second Meta-RaPS version was developed by utilizing a machine learning algorithm, Q Learning, which has been successfully applied to optimization problems whose output is a sequence of actions. In the third Meta-RaPS version, Path Relinking (PR) was implemented as a post-optimization method in which the new algorithm learns the good attributes by memorizing best solutions, and follows them to reach better solutions. The fourth proposed version of Meta-RaPS presented another form of learning with its ability to adaptively tune parameters. The efficiency of these approaches motivated us to redesign Meta-RaPS by removing the improvement phase and adding a more sophisticated Path Relinking method. The new Meta-RaPS could solve even the largest problems in much less time while keeping up the quality of its solutions. To evaluate their performance, all introduced versions were tested using the 0-1 Multidimensional Knapsack Problem (MKP). After comparing the proposed algorithms, Meta-RaPS PR and Meta-RaPS Q Learning appeared to be the algorithms with the best and worst performance, respectively. On the other hand, they could all show superior performance than other approaches to the 0-1 MKP in the literature

Randomized heuristics for the Capacitated Clustering Problem

In this paper, we investigate the adaptation of the Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the performance of these multi-start heuristic search methods when solving this NP-hard problem. The former is a memory-less approach that constructs independent solutions, while the latter is a memory-based method that constructs linked solutions, obtained by partially rebuilding previous ones. Both are based on the combination of greediness and randomization in the constructive process, and coupled with a subsequent local search phase. We propose these two multi-start methods and their hybridization and compare their performance on the CCP. Additionally, we propose a heuristic based on the mathematical programming formulation of this problem, which constitutes a so-called matheuristic. We also implement a classical randomized method based on simulated annealing to complete the picture of randomized heuristics. Our extensive experimentation reveals that Iterated Greedy performs better than GRASP in this problem, and improved outcomes are obtained when both methods are hybridized and coupled with the matheuristic. In fact, the hybridization is able to outperform the best approaches previously published for the CCP. This study shows that memory-based construction is an effective mechanism within multi-start heuristic search techniques