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Metaheuristics in Combinatorial Optimization: Overview and ...

¤¤¤¤S¡ ¡s x 1 v ¢ 1 x £¢¨§©§¨§¨¢ n v ¢ n v £ i D i s satisfies all the constra**in**ts §¢ ¦¥ ¦¥**Metaheuristics** **in** Comb**in**atorial **Optimization**:**Overview** **and** Conceptual ComparisonChristian BlumUniversité Libre de BruxellesAndrea RoliUniversità degli Studi di Bologna(Prepr**in**t)AbstractThe field of metaheuristics for the application to comb**in**atorial optimization problems is a rapidlygrow**in**g field of research. This is due to the importance of comb**in**atorial optimization problems for thescientific as well as the **in**dustrial world. We give a survey of the nowadays most important metaheuristicsfrom a conceptual po**in**t of view. We outl**in**e the different components **and** concepts that are used **in**the different metaheuristics **in** order to analyze their similarities **and** differences. Two very importantconcepts **in** metaheuristics are **in**tensification **and** diversification. These are the two forces that largelydeterm**in**e the behaviour of a metaheuristic. They are **in** some way contrary but also complementary toeach other. We **in**troduce a framework, that we call the I&D frame, **in** order to put different **in**tensification**and** diversification components **in**to relation with each other. Outl**in****in**g the advantages **and** disadvantagesof different metaheuristic approaches we conclude by po**in**t**in**g out the importance of hybridization ofmetaheuristics as well as the **in**tegration of metaheuristics **and** other methods for optimization.1 IntroductionMany optimization problems of both practical **and** theoretical importance concern themselves with thechoice of a “best” configuration of a set of variables to achieve some goals. They seem to divide naturally**in**to two categories: those where solutions are encoded with real-valued variables, **and** those wheresolutions are encoded with discrete variables. Among the latter ones we f**in**d a class of problems calledComb**in**atorial **Optimization** (CO) problems. Accord**in**g to [126], **in** CO problems, we are look**in**g for anobject from a f**in**ite – or possibly countably **in**f**in**ite – set. This object is typically an **in**teger number, asubset, a permutation, or a graph structure.¡Def**in**ition 1.1. A Comb**in**atorial **Optimization** problem ¢ P £ S f can be def**in**ed by:a set of variables X ¦¥ x 1 ¢¨§©§©§¢ x n ;variable doma**in**s D 1 ¢¨§©§¨§¨¢ D n ;constra**in**ts among variables;an objective function f to be m**in**imized, 1 where f : D 1 §¨§©§ D n IR ;The set of all possible feasible assignments isS is usually called a search (or solution) space, as each element of the set can be seen as a c**and**idatesolution. To solve a comb**in**atorial optimization problem one has to f**in**d s a solution S with m**in**imumobjective function value, that is, s£ f f s£ s S. s is called a globally optimal solution of S ¡ ¢ f £ **and**the set S ¡ S is called the set of globally optimal solutions.¡ 1 As maximiz**in**g an objective function f is the same as m**in**imiz**in**g f , **in** this work we will deal, without loss of generality, withm**in**imization problems.1

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