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Development of metaheuristic algorithms for the solution of nonlinear programming problems and complex compinatorial optimization problems

By Αθανάσιος Νικολακόπουλος

Abstract

Mathematical optimization is of primary importance for all practical sciences. However, the determination of the optimal solution in for many optimization problems is not possible with the conventional methodologies. This happens for example for the non linear programming problems that are non convex or the mathematical relations that compose them are non differentiable or not continuous. It also happens to combinatorial optimization problems characterized as NP-Hard. In the recent years, a new generation of heuristic - metaheuristic techniques has lead to the approximate solution of such complex optimization problems. The success of these methodologies relies also greatly to the progress of computer systems. This new research area is open and can be subjected to many improvements concerning the quality and the reproducibility of the solutions as well as the computational times. In the present thesis new hybrid metaheuristic algorithms have been developed for the solution of nonlinear and complex combinatorial optimisation problems. We give a concise presentation of the mathematical modelling of the problems under consideration and of the tools used for the recognition and analysis of their difficulty, which justifies the use of heuristic methods. First, a complete method for the solution of constrained nonlinear optimisation problems (NLP) has been developed. The method was based on the differential evolutionary algorithm with an ordered classification of the solutions (Line-Up Differential Evolution, shortly LUDE), which we propose for the solution of unconstrained nonlinear problems. The handling of linear and nonlinear constrains in the present work is done by incorporating them in an augmented Lagrange function. In this augmented function the values of the penalty parameters and the multipliers are adjusted, as the execution of the algorithm develops. The LUDE algorithm maintains a population of solutions, which is continuously improved as it passes from generation to generation. In the population of each generation the solutions are classified in an order that depends on the value of objective function produced from each one of them. The position of each solution in the line-up is very important, since it determines to what extent the crossover and mutation operators are applied to each solution. The efficiency of the proposed method is illustrated by solving numerous unconstrained and constrained optimization problems and comparing it with other optimization techniques that can be found in the literature. Τhe proposed method can face a larger variety of problems and in the majority of the cases it produces results of better quality with smaller computational cost and at the same time it presents great performance stability.

Topics: Αλγόριθμοι, Εξελικτικοί, Μη γραμμικός προγραμματισμός, Πρόβλημα περιοδεύοντος πωλητή, Αλγόριθμοι, Μεταευρετικοί, Συνδυαστική βελτιστοποίηση, Κατάταξη με υβριδοποίηση, Πρόβλημα δρομολόγησης στόλου οχημάτων, Αντίστροφα logistics, Evolutionary algorithms, Nonlinear programming, Traveling salesman problem, Metaheuristics algorithms, Combinatorial optimization, Sequencing by hybridization, Vehicle routing problem, Reverse logistics
Publisher: National Technical University of Athens (NTUA)
Year: 2006
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