Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

New variants of variable neighbourhood search for 0-1 mixed integer programming and clustering

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Many real-world optimisation problems are discrete in nature. Although recent rapid developments in computer technologies are steadily increasing the speed of computations, the size of an instance of a hard discrete optimisation problem solvable in prescribed time does not increase linearly with the computer speed. This calls for the development of new solution methodologies for solving larger instances in shorter time. Furthermore, large instances of discrete optimisation problems are normally impossible to solve to optimality within a reasonable computational time/space and can only be tackled with a heuristic approach. In this thesis the development of so called matheuristics, the heuristics which are based on the mathematical formulation of the problem, is studied and employed within the variable neighbourhood search framework. Some new variants of the variable neighbourhood searchmetaheuristic itself are suggested, which naturally emerge from exploiting the information from the mathematical programming formulation of the problem. However, those variants may also be applied to problems described by the combinatorial formulation. A unifying perspective on modern advances in local search-based metaheuristics, a so called hyper-reactive approach, is also proposed. Two NP-hard discrete optimisation problems are considered: 0-1 mixed integer programming and clustering with application to colour image quantisation. Several new heuristics for 0-1 mixed integer programming problem are developed, based on the principle of variable neighbourhood search. One set of proposed heuristics consists of improvement heuristics, which attempt to find high-quality near-optimal solutions starting from a given feasible solution. Another set consists of constructive heuristics, which attempt to find initial feasible solutions for 0-1 mixed integer programs. Finally, some variable neighbourhood search based clustering techniques are applied for solving the colour image quantisation problem. All new methods presented are compared to other algorithms recommended in literature and a comprehensive performance analysis is provided. Computational results show that the methods proposed either outperform the existing state-of-the-art methods for the problems observed, or provide comparable results. The theory and algorithms presented in this thesis indicate that hybridisation of the CPLEX MIP solver and the VNS metaheuristic can be very effective for solving large instances of the 0-1 mixed integer programming problem. More generally, the results presented in this thesis suggest that hybridisation of exact (commercial) integer programming solvers and some metaheuristic methods is of high interest and such combinations deserve further practical and theoretical investigation. Results also show that VNS can be successfully applied to solving a colour image quantisation problem.Support from the Mathematical Institute, Serbian Academy of Sciences and Arts, are acknowledged for this research

Contribution to the development of efficient algorithms for solving complex single-objective and multi-objective optimization models

L’optimització en enginyeria de processos és un àrea molt estesa que ha anat evolucionant al llarg del temps i ha passat de ser una metodologia d'interès purament acadèmic a una tecnologia que té, i que contínua tenint, gran impacte en la indústria. En aquesta tesi ens hem centrat en el desenvolupament mètodes basats en dues eines típiques d'optimització: programació matemàtica i metaheurístiques. Els objectius d'aquesta tesi són: el primer és desenvolupar una metaheuristica híbrida per a l'optimització del disseny de cadenes de subministrament, d'un sol objectiu (cost o benefici), on tots els paràmetres són coneguts a priori; el segon és desenvolupar un algorisme efectiu per a reducció d'objectius facilitant la resolució de problemes multi-objectiu; i finalment s'han implementat una sèrie de millores en el mètode de la restricció èpsilon per millorar l'eficiència en la resolució de problemes multi-objectiu. Tots els algorismes presentats han estat comparats i avaluats amb els mètodes establerts per la literatura.La optimización en ingeniería de procesos es un área muy extensa que ha ido evolucionando a lo largo del tiempo y ha pasado de ser una metodología de interés puramente académico a una tecnología que tiene, y que continua teniendo, gran impacto en la industria. En esta tesis nos hemos centrado en el desarrollo de métodos basados en dos herramientas típicas de optimización: programación matemática y metaheurísticas. Los objetivos de esta tesis son: el primero es desarrollar una metaheuristica híbrida para la optimización del diseño de cadenas de suministro, de un solo objetivo (coste o beneficio), donde todos los parámetros son conocidos a priori; el segundo es desarrollar un algoritmo efectivo para la reducción de objetivos facilitando la resolución de problemas multi-objetivo; y finalmente se han implementado una serie de mejoras en el método de la restricción epsilon para mejorar la eficiencia en la resolución de problemas multi-objetivo. Todos los algoritmos presentados han sido comparados y evaluados con los métodos establecidos por la literatura.Optimization has become a major area in process systems engineering. It has evolved from a methodology of academic interest into a technology that has and continues to make significant impact in industry. In this thesis we have focused on development of tools based on two standard optimization methods: mathematical programming and metaheuristics. The objectives of this thesis are: firstly, the development of a hybrid metaheuristic for optimizing the design of supply chains, single objective (cost or benefit), where all parameters are known previously; secondly, the development of an effective algorithm for objective reduction facilitating the resolution of multi-objective problems; and finally, we improved the epsilon-constraint algorithm in multi-objective optimization. All the algorithms presented have been assessed with the methods established in the literature

Model-Based Heuristics for Combinatorial Optimization

Many problems arising in several and different areas of human knowledge share the characteristic of being intractable in real cases. The relevance of the solution of these problems, linked to their domain of action, has given birth to many frameworks of algorithms for solving them. Traditional solution paradigms are represented by exact and heuristic algorithms. In order to overcome limitations of both approaches and obtain better performances, tailored combinations of exact and heuristic methods have been studied, giving birth to a new paradigm for solving hard combinatorial optimization problems, constituted by model-based metaheuristics. In the present thesis, we deepen the issue of model-based metaheuristics, and present some methods, belonging to this class, applied to the solution of combinatorial optimization problems

New variants of variable neighbourhood search for 0-1 mixed integer programming and clustering

Many real-world optimisation problems are discrete in nature. Although recent rapid developments in computer technologies are steadily increasing the speed of computations, the size of an instance of a hard discrete optimisation problem solvable in prescribed time does not increase linearly with the computer speed. This calls for the development of new solution methodologies for solving larger instances in shorter time. Furthermore, large instances of discrete optimisation problems are normally impossible to solve to optimality within a reasonable computational time/space and can only be tackled with a heuristic approach. In this thesis the development of so called matheuristics, the heuristics which are based on the mathematical formulation of the problem, is studied and employed within the variable neighbourhood search framework. Some new variants of the variable neighbourhood searchmetaheuristic itself are suggested, which naturally emerge from exploiting the information from the mathematical programming formulation of the problem. However, those variants may also be applied to problems described by the combinatorial formulation. A unifying perspective on modern advances in local search-based metaheuristics, a so called hyper-reactive approach, is also proposed. Two NP-hard discrete optimisation problems are considered: 0-1 mixed integer programming and clustering with application to colour image quantisation. Several new heuristics for 0-1 mixed integer programming problem are developed, based on the principle of variable neighbourhood search. One set of proposed heuristics consists of improvement heuristics, which attempt to find high-quality near-optimal solutions starting from a given feasible solution. Another set consists of constructive heuristics, which attempt to find initial feasible solutions for 0-1 mixed integer programs. Finally, some variable neighbourhood search based clustering techniques are applied for solving the colour image quantisation problem. All new methods presented are compared to other algorithms recommended in literature and a comprehensive performance analysis is provided. Computational results show that the methods proposed either outperform the existing state-of-the-art methods for the problems observed, or provide comparable results. The theory and algorithms presented in this thesis indicate that hybridisation of the CPLEX MIP solver and the VNS metaheuristic can be very effective for solving large instances of the 0-1 mixed integer programming problem. More generally, the results presented in this thesis suggest that hybridisation of exact (commercial) integer programming solvers and some metaheuristic methods is of high interest and such combinations deserve further practical and theoretical investigation. Results also show that VNS can be successfully applied to solving a colour image quantisation problem.EThOS - Electronic Theses Online ServiceMathematical Institute, Serbian Academy of Sciences and ArtsGBUnited Kingdo