Penalização iterativa da Evolução Diferencial em otimização multimodal

Abstract

Mathematical formulations of real case studies take into account the maximum amount of information available on the analyzed case. Due to this fact, the formulations generally present characteristics such as non-linearity, discontinuity and high complexity. From the context in which these formulations are generated, they can be transformed into optimization problems by creating a merit function, commonly called the objective function, which can be minimization or maximization problems. This class of problems may present a high number of global minimum points, due to this fact, they are called multimodal function. Due to the multimodal character of these functions, multipopulation methods have been employed in order to obtain the highest number of points of global minimum. In the present work, two new approaches are developed that use an iterative penalization technique and a multipopulation procedure together with the Differential Evolution algorithm in order to obtain the set of solutions for multimodal optimization problems. The first method proposed is the Multipopulation Differential Evolution with iterative technique of modification of the objective function, EDM–TIMF, and the second method proposed is the Differential Evolution with Initialization, EDI. In this second proposal, the EDM–TIMF method is used a startup tool of the initial populations and from a given moment the Differential Evolution is used to solve the problem addressed. In these approaches, subpopulations evolve simultaneously throughout the iterative process. Simulations are carried out in order to study the qualitative behavior of the methods developed from the sensitivity analysis of the parameters inherent to the methods. Subsequently, the ED, EDM–TIMF and EDI methods are applied to evaluate them, and validate their capability in solving multimodal problems, in a set with 10 multimodal benchmark functions and in two engineering problems, the first in the area phase balance in mixtures and the second in the area of robotics. However, the ED method is not applied to the problem of robotics. The most complex task for the developed methods is the application of them in the problem of Inverse Robotic Kinematics due to the mathematical formulation being in the R8 and has a high number of solutions in the interval [−1,1]. Based on the results obtained, it is concluded that all methods are suitable for these tasks. The EDI method presented subtle superiority in obtaining a greater number of distinct global minimum points compared to the EDM–TIMF method in all multimodal benchmark functions and in the real problems used in this thesis.