Influence of Weighting factor and Crossover constant on the behavior of Differential Evolution Algorithms with a Penalty Function approach

Abstract

Differential Evolution (DE) is one of the most popular evolutionary optimization technique on continuous domains based on simplicity, effectiveness and robustness. The weighting factor(F) and crossover constant(CR) allows the construction of a new trial element based on the current and mutant elements. The crossover constant controls which and how many components are mutated in each element of the current population. The work in the present paper aims to analyze the impact the weighting factor and the crossover constant, has on the behavior of DE. The influence of the crossover constant on the distribution of the number of mutated components and on the probability for a component to be taken from mutant vector (mutation probability) is analyzed for several variants of weighting factor and crossover factor, including classical binomial and exponential strategies. For each weighting and crossover variant the relationship between the crossover and mutation probability is identified and its impact on the choice and adaptation of control parameters is analyzed numerically and graphically. Ten different strategies (variations) of DE with penalty function approach are analyzed with various population sizes, crossover and weighting factors and applied to the problem of minimizing the cost of the active parts of the power objects. Constraints resulting from international specifications are taken into account. The Objective functions that are optimized are minimizations dependent on multiple input variables. All constraints are normalized and modeled as inequalities.