Constraint-Handling in Evolutionary Optimization

This work introduces a hybrid PSO algorithm which includes perturbation operators to keep population diversity. A new neighborhood structure for Particle Swarm Optimization called Singly-Linked Ring is implemented. The approach proposes a neighborhood similar to the ring structure, but which has an innovative neighbors selection. The objective is to avoid the premature convergence into local optimum. A special technique to handle equality constraints with low side effects on the diversity is the main feature of this contribution. Two perturbation operators are used to improve the exploration, applying the modification only in the particle best population.We show through a number of experiments how, by keeping the selection pressure on a decreasing fraction of the population, COPSO can consistently solve a benchmark of constrained optimization problems.

The necessity of approaching the boundary between the feasible and infeasible search space for many constrained optimization problems is a paramount challenge for every constraint-handling technique. It is true that many of the stateof- the-art constraint-handling techniques performs well when facing constrained problems. However, it is a common situation that reaching the boundary between the feasible and infeasible search space could be a difficult task for some particular problems. Firstly, this chapter shows a general overview of the constraint-handling techniques based on a boundary approach and emphasizing a recent proposal applying a more general boundary operator. In addition, the chapter includes some particular considerations related to the implementation aspects of the boundary approach when facing problems with one o more constraints. Another important issue also considered here is about the implementation of this approach when taking into account different search engines. On this direction, some basic examples are depicted as guidelines for possible implementations under well-known metaheuristics as Evolutionary Algorithms (EAs), Particle Swarm Optimization (PSO), and Ant ColonyOptimization (ACO). To validate the boundary approach implemented under the above metaheuristics, an experimental study is presented in which well-known problems were considered. Finally, a brief summary of the chapter and some ideas for future works are given which could help the researchers interested in developing advanced constraint-handling techniques.

While research on constrained optimization using evolutionary algorithms has been actively pursued, it has had to face the problem that the ability to solve multi-modal problems is insufficient, that the ability to solve problems with equality constraints is inadequate, and that the stability and efficiency of searches is low. We have proposed the εDE, defined by applying the ε constrained method to differential evolution (DE). It is shown that the εDE is a fast and stable algorithm that is robust to multi-modal problems and it can solve problems with many equality constraints by introducing a gradient-based mutation which finds a feasible point using the gradient of constraints. In this chapter, an improved εDE is proposed, in which faster reduction of the relaxation of equality constraints in the ε constrained method and higher gradient-based mutation rate are adopted in order to solve problems with many equality constraints and to find feasible solutions faster and very stably. Also, cutting off and reflecting back solutions outside of search space are adopted to improve the efficiency in finding optimal solutions. The improved εDE realizes stable and efficient searches, and can solve difficult constrained optimization problems with equality constraints. The advantage of the improved εDE is shown by applying it to twenty four constrained problems of various types.

In this Chapter we present the modification of a Differential Evolution algorithm to solve constrained optimization problems. The changes include a deterministic and a self-adaptive parameter control in two of the Differential Evolution parameters and also in two parameters related with the constraint-handling mechanism. The proposed approach is extensively tested by using a set of well-known test problems and performance measures found in the specialized literature. Besides analyzing the final results obtained by the algorithm with respect to its original version, some interesting findings regarding the behavior found in the approach and in the values observed on each of the parameters controlled are also discussed.

This chapter proposes a constraint handling technique for multi-objective evolutionary algorithms based on an adaptive penalty function and a distance measure. These two functions vary dependent upon the objective function value and the sum of constraint violations of an individual. Through this design, the objective space is modified to account for the performance and constraint violation of each individual. The modified objective functions are used in the non-dominance sorting to facilitate the search of optimal solutions not only in the feasible space but also in the infeasible regions. The search in the infeasible space is designed to exploit those individuals with better objective values and lower constraint violations. The number of feasible individuals in the population is used to guide the search process either toward finding more feasible solutions or favor in search for optimal solutions. The proposed method is simple to implement and does not need any parameter tuning. The constraint handling technique is tested on several constrained multi-objective optimization problems and has shown superior results compared to some chosen state-of-the-art designs.

Real life optimization problems often involve one or more constraints and in most cases, the optimal solutions to such problems lie on constraint boundaries. The performance of an optimization algorithm is known to be largely dependent on the underlying mechanism of constraint handling. Most population based stochastic optimization methods prefer a feasible solution over an infeasible solution during their course of search. Such a preference drives the population to feasibility first before improving its objective function value which effectively means that the solutions approach the constraint boundaries from the feasible side of the search space. In this chapter, we introduce an evolutionary algorithm that explicitly maintains a small percentage of infeasible solutions close to the constraint boundaries during its course of evolution. The presence of marginally infeasible solutions in the population allows the algorithm to approach the constraint boundary from the infeasible side of the search space in addition to its approach from the feasible side of the search space via evolution of feasible solutions. Furthermore, “good” infeasible solutions are ranked higher than the feasible solutions, thereby focusing the search for the optimal solutions near the constraint boundaries. The performance of the proposed algorithm is compared with Non-dominated Sorting Genetic Algorithm II (NSGA-II) on a set of single and multi-objective test problems. The results clearly indicate that the rate of convergence of the proposed algorithm is better than NSGA-II on the studied test problems. Additionally, the algorithm provides a set of marginally infeasible solutions which are of great use in trade-off studies.

A genetic algorithm (GA) is hybridized with an artificial immune system (AIS) as an alternative to tackle constrained optimization problems in engineering. The AIS is inspired by the clonal selection principle and is embedded into a standard GA search engine in order to help move the population into the feasible region. The resulting GA-AIS hybrid is tested in a suite of constrained optimization problems with continuous variables, as well as structural and mixed integer reliability engineering optimization problems. In order to improve the diversity of the population, a variant of the algorithm is developed with the inclusion of a clearing procedure. The performance of the GA-AIS hybrids is compared with that of alternative techniques, such as the Adaptive Penalty Method, and the Stochastic Ranking technique, which represent two different types of constraint handling techniques that have been shown to provide good results in the literature.

An aspect that often causes difficulties when using Genetic Algorithms for optimization is that these algorithms operate as unconstrained search procedures and most of the real-world problems have constraints of different types. There is a lack of efficient constraint handling technique to bias the search in constrained search spaces toward the feasible regions. We propose a novel methodology to be coupled with a Genetic Algorithm to solve optimization problems with inequality constraints. This methodology can be seen as a local search operator that uses quadratic and linear approximations for both objective function and constraints. In the local search phase, these approximations define an associated problem with a quadratic objective function and quadratic and/or linear constraints that is solved using an LMI (linear matrix inequality) formulation. The solution of this associated problems is then re-introduced in the GA population.We test the proposed methodology with a set of analytical function and the results show that the hybrid algorithm has a better performancewhen compared to the same Genetic Algorithmwithout the proposed local search operator. The tests also suggest that the proposed methodology is at least equivalent, and sometimes better than other methods that have been reported recently in literature.

Constraint-handling techniques for evolutionary multiobjective aerodynamic and multidisciplinary designs are focused. Because number of evaluations is strictly limited in aerodynamic or multidisciplinary design optimization due to expensive computational fluid dynamics (CFD) simulations for aerodynamic evaluations, very efficient and robust constraint-handling technique is required for aerodynamic and multidisciplinary design optimizations. First, in Section 2, features of aerodynamic design optimization problems are discussed. Then, in Section 3 constraint-handling techniques used for aerodynamic and multidisciplinary designs are overviewed. Then, an efficient constraint-handling technique suitable to aerodynamic and multidisciplinary designs is introduced with real-world aerodynamic and multidisciplinary applications. Finally, an efficient geometry-constraint-handling technique commonly used for aerodynamic design optimizations is presented.

Artificial Immune Systems (AIS) are computational intelligent systems inspired by some processes or theories observed in the biological immune system. They have been applied to solve a wide range of machine learning and optimization problems. In this chapter the main AIS-based proposals for solving constrained numerical optimization problems are shown. Although the first works were hybrid solutions partially based on Genetic Algorithms, the most recent proposals are algorithms completely based on immune features.We show that these algorithms represent viable alternatives to the penalty functions and other similar mechanisms to handle constraints in numerical optimization problems.