Operadores para algoritmos genéticos baseados em aproximações quadráticas de funções de variáveis contínuas

Abstract

This thesis investigates the possibility of using quadratic approximations of functions with the purpose of building new operators for genetic algorithms, applied to the optimization of continuous variable functions. The basic formulation employed in all cases is the employment of the set of samples of objective functions and constraint functions that is already obtained through the execution of the typical operations of genetic algorithms. With this set of samples, the quadratic approximations of the several functions are calculated and, as the genetic algorithm goes getting new samples, such approximations are updated. This thesis proposes, upon such approximations: (i) an operator that performs the coordinate correction of the variable space; (ii) an operator that generates estimates of the optimum for mono-objective problems with a single constraint; (iii) an operator that generates estimates of the optimum for mono-objective problems with several inequality constraints; (iv) an operator that generates locally refined estimates of the Pareto-set points of unconstrained multiobjective problems; and (v) an operator that generates locally refined estimates of the Pareto-set points of multiobjective problems with multiple inequality constraints. The three last operators are built on the basis of Linear Matrix Inequality (LMI) formulations. As a by-product of this thesis, a new metric is proposed here, for the purpose of comparing the performances of multiobjective optimization algorithms in the task of generating representative sample sets for the Pareto-optimal sets of multiobjective problems: the {em sphere-counting} metric. The results obtained suggest that all proposed operators are capable of leading to significant enhancements in the convergence rate, in the proportion of convergence and in the solution precision. Additional studies are necessary, in the case of the multiobjective operators, for enhancing the extension of the Pareto-estimate sets that are obtained.