Aperfeiçoamentos em métodos de otimização sem derivadas

Abstract

This work proposes improvements for some deterministic derivative-free optimization methods, in particular for line search optimization methods and the Nelder-Mead simplex method. Regarding the first contribution, this work proposes a line search optimization framework based on the (v, a)-patterns, that are used for the multimodality characterization of nonlinear functions. Based on this, the framework can map multiple minima throughout successive interval breakdowns whenever a local maximum is detected. The proposed framework is coupled with the golden section method, originating a novel linear search optimization method called multimodal golden section, which inherits the convergence properties of the underlying method. Numerical experiments depict the multimodal feature of the framework. Regarding the second contribution, this paper proposes the use of a lexicographic operator to deal with box and inequality constraints for the classic Nelder-Mead simplex method. Also, a new initial simplex initialization strategy is proposed to prevent premature degeneration. The proposed modifications do not alter the original structure of the algorithm. Experiments are conducted and the results are compared with traditional simplex initializations and constraint handling strategies, demonstrating the main characteristics of the contribution.