Tetsuyuki Takahama

Constrained Optimization by the ε Constrained Differential Evolution with Gradient-Based Mutation and Feasible Elites

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, which have many local solutions within a feasible region, is insufficient, that the ability to solve problems with equality constraints is inadequate, and that the stability and efficiency of searches is low. We proposed the epsivDE, defined by applying the epsiv constrained method to a differential evolution (DE). DE is a simple, fast and stable population based search algorithm that is robust to multi-modal problems. The epsivDE is improved to solve problems with many equality constraints by introducing a gradient-based mutation that finds feasible point using the gradient of constraints at an infeasible point. Also the epsivDE is improved to find feasible solutions faster by introducing elitism where more feasible points are preserved as feasible elites. The improved epsivDE realizes stable and efficient searches that can solve multi-modal problems and those with equality constraints. The advantage of the epsivDE is shown by applying it to twenty four constrained problems of various types.

Constrained optimization by applying the /spl alpha/ constrained method to the nonlinear simplex method with mutations

Constrained optimization problems are very important and frequently appear in the real world. The /spl alpha/ constrained method is a new transformation method for constrained optimization. In this method, a satisfaction level for the constraints is introduced, which indicates how well a search point satisfies the constraints. The /spl alpha/ level comparison, which compares search points based on their level of satisfaction of the constraints, is also introduced. The /spl alpha/ constrained method can convert an algorithm for unconstrained problems into an algorithm for constrained problems by replacing ordinary comparisons with the /spl alpha/ level comparisons. In this paper, we introduce some improvements including mutations to the nonlinear simplex method to search around the boundary of the feasible region and to control the convergence speed of the method, we apply the /spl alpha/ constrained method and we propose the improved /spl alpha/ constrained simplex method for constrained optimization problems. The effectiveness of the /spl alpha/ constrained simplex method is shown by comparing its performance with that of the stochastic ranking method on various constrained problems.

Constrained optimization by the ε constrained differential evolution with an archive and gradient-based mutation

The ε constrained method is an algorithm transformation method, which can convert algorithms for unconstrained problems to algorithms for constrained problems using the ε level comparison, which compares search points based on the pair of objective value and constraint violation of them. We have proposed the ε constrained differential evolution (εDE), which is the combination of the ε constrained method and differential evolution (DE). It has been shown that the εDE can run very fast and can find very high quality solutions. Also, we proposed the εDE with gradient-based mutation (εDEg), which utilized gradients of constraints in order to solve problems with difficult constraints. In this study, we propose the ε constrained DE with an archive and gradient-based mutation (εDEag). The εDEag utilizes an archive to maintain the diversity of individuals and adopts a new way of selecting the ε level control parameter in the εDEg. The 18 problems, which are given in special session on “Single Objective Constrained RealParameter Optimization” in CEC2010, are solved by the εDEag and the results are shown in this paper.

Constrained optimization by the ε constrained hybrid algorithm of particle swarm optimization and genetic algorithm

The e constrained method is an algorithm transformation method, which can convert algorithms for unconstrained problems to algorithms for constrained problems using the e level comparison that compares search points based on the constraint violation of them. We proposed the e constrained particle swarm optimizer ePSO, which is the combination of the e constrained method and particle swarm optimization. The ePSO can run very fast and find very high quality solutions, but the ePSO is not very stable and sometimes can only find lower quality solutions. On the contrary, the eGA, which is the combination of the e constrained method and GA, is very stable and can find high quality solutions, but it is difficult for the eGA to find higher quality solutions than the ePSO. In this study, we propose the hybrid algorithm of the ePSO and the eGA to find very high quality solutions stably. The effectiveness of the hybrid algorithm is shown by comparing it with various methods on well known nonlinear constrained problems.

Constrained Optimization by ε Constrained Particle Swarm Optimizer with ε-level Control

In this study, e constrained particle swarm optimizer ePSO, which is the combination of the e constrained method and particle swarm optimization, is proposed to solve constrained optimization problems. The e constrained methods can convert algorithms for unconstrained problems to algorithms for constrained problems using the e level comparison, which compares the search points based on the constraint violation of them. In the e PSO, the agents who satisfy the constraints move to optimize the objective function and the agents who don’t satisfy the constraints move to satisfy the constraints. Also, the way of controlling e-level is given to solve problems with equality constraints. The effectiveness of the e PSO is shown by comparing the e PSO with GENOCOP5.0 on some nonlinear constrained problems with equality constraints.

Efficient constrained optimization by the ε constrained adaptive differential evolution

The ε constrained method is an algorithm transformation method, which can convert algorithms for unconstrained problems to algorithms for constrained problems using the ε level comparison, which compares search points based on the pair of objective value and constraint violation of them. We have proposed the ε constrained differential evolution εDE, which is the combination of the ε constrained method and differential evolution (DE), and have shown that the εDE can run very fast and can find very high quality solutions. In this study, we propose the ε constrained adaptive DE (εADE), which adopts a new and stable way of controlling the ε level and adaptive control of algorithm parameters in DE. The εADE is very efficient constrained optimization algorithm that can find high-quality solutions in very small number of function evaluations. It is shown that the εADE can find near optimal solutions stably in about half the number of function evaluations compared with various other methods on well known nonlinear constrained problems.

Solving Nonlinear Constrained Optimization Problems by the ε Constrained Differential Evolution

The ε constrained method is an algorithm transformation method, which can convert algorithms for unconstrained problems to algorithms for constrained problems using the ε level comparison that compares search points based on the constraint violation of them. We propose the ε constrained differential evolution εDE, which is the combination of the ε constrained method and differential evolution (DE). DE is a simple, fast and stable search algorithm that is robust to multi-modal problems. It is expected that the εDE is robust to multi-modal problems, can run very fast and can find very high quality solutions. The effectiveness of the εDE is shown by comparing it with various methods on well known nonlinear constrained problems.

Solving Difficult Constrained Optimization Problems by the ε Constrained Differential Evolution with Gradient-Based Mutation

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 eDE, defined by applying the e constrained method to differential evolution (DE). It is shown that the eDE 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 eDE is proposed, in which faster reduction of the relaxation of equality constraints in the e 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 eDE realizes stable and efficient searches, and can solve difficult constrained optimization problems with equality constraints. The advantage of the improved eDE is shown by applying it to twenty four constrained problems of various types.

Constrained Optimization by ε Constrained Differential Evolution with Dynamic ε-Level Control

In this chapter, the improved e constrained differential evolution (eDE) is proposed to solve constrained optimization problems with very small feasible region, such as problems with equality constraints, efficiently. The eDE is the combination of the e constrained method and differential evolution. In general, it is very difficult to solve constrained problems with very small feasible region. To solve such problems, static control schema of allowable constraint violation is often used, where solutions are searched within enlarged region specified by the allowable violation and the region is reduced to the feasible region gradually. However, the proper control depends on the initial population and searching process. In this study, the dynamic control of allowable violation is proposed to solve problems with equality constraints efficiently. In the eDE, the amount of allowable violation can be specified by the e-level. The effectiveness of the eDE with dynamic e-level control is shown by comparing with the original eDE and well known optimization method on some nonlinear constrained problems with equality constraints.

Efficient Constrained Optimization by the ε Constrained Rank-Based Differential Evolution

The ε constrained method is an algorithm transformation method, which can convert algorithms for unconstrained problems to algorithms for constrained problems using the ε level comparison, which compares search points based on the pair of objective value and constraint violation of them. We have proposed the ε constrained differential evolution εDE, which is the combination of the ε constrained method and differential evolution (DE), and have shown that the εDE can run very fast and can find very high quality solutions. In this study, we propose the ε constrained rank-based DE (εRDE), which adopts a new and simple scheme of controlling algorithm parameters in DE. In the scheme, different parameter values are selected for each individual. Small scaling factor and large crossover rate are selected for good individuals to improve the efficiency of search. Large scaling factor and small crossover rate are selected for bad individuals to improve the stability of search. The goodness is given by the ranking information. The εRDE is a very efficient constrained optimization algorithm that can find high-quality solutions in very small number of function evaluations. It is shown that the εRDE can find near optimal solutions stably in about half the number of function evaluations compared with various other methods on well known nonlinear constrained problems.