Efrén Mezura-Montes

A simple multimembered evolution strategy to solve constrained optimization problems

This work presents a simple multimembered evolution strategy to solve global nonlinear optimization problems. The approach does not require the use of a penalty function. Instead, it uses a simple diversity mechanism based on allowing infeasible solutions to remain in the population. This technique helps the algorithm to find the global optimum despite reaching reasonably fast the feasible region of the search space. A simple feasibility-based comparison mechanism is used to guide the process toward the feasible region of the search space. Also, the initial stepsize of the evolution strategy is reduced in order to perform a finer search and a combined (discrete/intermediate) panmictic recombination technique improves its exploitation capabilities. The approach was tested with a well-known benchmark. The results obtained are very competitive when comparing the proposed approach against other state-of-the art techniques and its computational cost (measured by the number of fitness function evaluations) is lower than the cost required by the other techniques compared.

Constraint-handling in nature-inspired numerical optimization: Past, present and future

Abstract In their original versions, nature-inspired search algorithms such as evolutionary algorithms and those based on swarm intelligence, lack a mechanism to deal with the constraints of a numerical optimization problem. Nowadays, however, there exists a considerable amount of research devoted to design techniques for handling constraints within a nature-inspired algorithm. This paper presents an analysis of the most relevant types of constraint-handling techniques that have been adopted with nature-inspired algorithms. From them, the most popular approaches are analyzed in more detail. For each of them, some representative instantiations are further discussed. In the last part of the paper, some of the future trends in the area, which have been only scarcely explored, are briefly discussed and then the conclusions of this paper are presented.

Modified Differential Evolution for Constrained Optimization

In this paper, we present a Differential-Evolution based approach to solve constrained optimization problems. The aim of the approach is to increase the probability of each parent to generate a better offspring. This is done by allowing each solution to generate more than one offspring but using a different mutation operator which combines information of the best solution in the population and also information of the current parent to find new search directions. Three selection criteria based on feasibility are used to deal with the constraints of the problem and also a diversity mechanism is added to maintain infeasible solutions located in promising areas of the search space. The approach is tested in a set of test problems proposed for the special session on Constrained Real Parameter Optimization. The results obtained are discussed and some conclusions are established.

Differential evolution in constrained numerical optimization: An empirical study

Motivated by the recent success of diverse approaches based on differential evolution (DE) to solve constrained numerical optimization problems, in this paper, the performance of this novel evolutionary algorithm is evaluated. Three experiments are designed to study the behavior of different DE variants on a set of benchmark problems by using different performance measures proposed in the specialized literature. The first experiment analyzes the behavior of four DE variants in 24 test functions considering dimensionality and the type of constraints of the problem. The second experiment presents a more in-depth analysis on two DE variants by varying two parameters (the scale factor F and the population size NP), which control the convergence of the algorithm. From the results obtained, a simple but competitive combination of two DE variants is proposed and compared against state-of-the-art DE-based algorithms for constrained optimization in the third experiment. The study in this paper shows (1) important information about the behavior of DE in constrained search spaces and (2) the role of this knowledge in the correct combination of variants, based on their capabilities, to generate simple but competitive approaches.

Multi-objective Optimization Using Differential Evolution: A Survey of the State-of-the-Art

Differential Evolution is currently one of the most popular heuristics to solve single-objective optimization problems in continuous search spaces. Due to this success, its use has been extended to other types of problems, such as multi-objective optimization. In this chapter, we present a survey of algorithms based on differential evolution which have been used to solve multi-objective optimization problems. Their main features are described and, based precisely on them, we propose a taxonomy of approaches. Some theoretical work found in the specialized literature is also provided. To conclude, based on our findings, we suggest some topics that we consider to be promising paths for future research in this area.

An empirical study about the usefulness of evolution strategies to solve constrained optimization problems

In this paper, we explore the capabilities of different types of evolution strategies (ES) to solve global optimization problems with constraints. The aim is to highlight the idea that the selection of the search engine is more critical than the selection of the constraint-handling mechanism, which can be very simple indeed. We show how using just three simple comparison criteria based on feasibility, the simple evolution strategy can be led to the feasible region of the search space and find the global optimum solution (or a very good approximation of it). Different ES including a variation of a (μ+1) − ES and with or without correlated mutation were implemented. Such approaches were tested using a well-known test suite for constrained optimization. Furthermore, the most competitive version found (among those five) was compared against three state-of-the-art approaches and it was also compared against a GA using the same constraint-handling approach. Finally, our evolution strategy was used to solve some engineering design problems.

Multiobjective Evolutionary Algorithms in Aeronautical and Aerospace Engineering

Nowadays, the solution of multiobjective optimization problems in aeronautical and aerospace engineering has become a standard practice. These two fields offer highly complex search spaces with different sources of difficulty, which are amenable to the use of alternative search techniques such as metaheuristics, since they require little domain information to operate. From the several metaheuristics available, multiobjective evolutionary algorithms (MOEAs) have become particularly popular, mainly because of their availability, ease of use, and flexibility. This paper presents a taxonomy and a comprehensive review of applications of MOEAs in aeronautical and aerospace design problems. The review includes both the characteristics of the specific MOEA adopted in each case, as well as the features of the problems being solved with them. The advantages and disadvantages of each type of approach are also briefly addressed. We also provide a set of general guidelines for using and designing MOEAs for aeronautical and aerospace engineering problems. In the final part of the paper, we provide some potential paths for future research, which we consider promising within this area.

Promising infeasibility and multiple offspring incorporated to differential evolution for constrained optimization

In this paper, we incorporate a diversity mechanism to the differential evolution algorithm to solve constrained optimization problems without using a penalty function. The aim is twofold: (1) to allow infeasible solutions with a promising value of the objective function to remain in the population and also (2) to increase the probabilities of an individual to generate a better offspring while promoting collaboration of all the population to generate better solutions. These goals are achieved by allowing each parent to generate more than one offspring. The best offspring is selected using a comparison mechanism based on feasibility and this child is compared against its parent. To maintain diversity, the proposed approach uses a mechanism successfully adopted with other evolutionary algorithms where, based on a parameter Sr a solution (between the best offspring and the current parent) with a better value of the objective function can remain in the population, regardless of its feasibility. The proposed approach is validated using test functions from a well-known benchmark commonly adopted to validate constraint-handling techniques used with evolutionary algorithms. The statistical results obtained by the proposed approach are highly competitive (based on quality, robustness and number of evaluations of the objective function) with respect to other constraint-handling techniques, either based on differential evolution or on other evolutionary algorithms, that are representative of the state-of-the-art in the area. Finally, a small set of experiments were made to detect sensitivity of the approach to its parameters.

Constrained Optimization via Multiobjective Evolutionary Algorithms

In this chapter, we present a survey of constraint-handling techniques based on evolutionary multiobjective optimization concepts. We present some basic definitions required to make this chapter self-contained, and then introduce the way in which a global (single-objective) nonlinear optimization problem is transformed into an unconstrained multiobjective optimization problem. A taxonomy of methods is also proposed and each of them is briefly described. Some interesting findings regarding common features of the approaches analyzed are also discussed.

Multiple trial vectors in differential evolution for engineering design

This article presents a modified version of the differential evolution algorithm to solve engineering design problems. The aim is to allow each parent vector in the population to generate more than one trial (child) vector at each generation and therefore to increase its probability of generating a better one. To deal with constraints, some criteria based on feasibility and a diversity mechanism to maintain infeasible solutions in the population are used. The approach is tested on a set of well-known benchmark problems. After that, it is used to solve engineering design problems and its performance is compared with those provided by typical penalty function approaches and also against state-of-the-art techniques.