Variants of the Epsilon Constrained Method in a Memetic Differential Evolution: A Comparative Study

Memetic approaches are composed of three general processes, a global optimizer, a set of local-search operators, and a coordination mechanism; which are defined depending on the problem to be optimized. For constrained optimization problems (COPs), memetic algorithms require the incorporation of a constraint handler that guides the search to the feasible regions of the search space. In this regard, the epsilon-constrained method has demonstrated to operate correctly in memetic approaches by transforming a COP into an unconstrained problem during a certain period of the search process. This constraint handler uses a tolerance level that promotes the exploration, mainly in COPs where there are disjoint feasible regions or equality constraints. Nevertheless, epsilon-constrained depends on a set of parameters that determine its behavior, so five variants have emerged in the control of its tolerance, (1) static, (2) dynamic, (3) truncated, (4) threshold, and (5) adaptive. This study focuses on determining the most appropriate control technique in a memetic approach and its relation to the performance and final results of the algorithm. For the study, a memetic differential evolution (MDE) is implemented, whose coordination mechanism controls the activation of three local search methods. Each epsilon-level control mechanism is incorporated separately within the MDE and is tested in eighteen well-known test problems. The results suggest that there is a benefit through the use of adaptive/dynamic mechanisms while reducing the budget for fitness evaluations. Likewise, its advantage is exhibited in functions with non-separable equality constraints. Finally, results determine that there is no benefit relationship between how to control the epsilon-level and the performance of the local optimizers used in this study.