A Diversity Promotion Study in Constrained Optimizations

Evolutionary Algorithms have been widely used to solve difficult constrained optimization problems. However, Evolutionary algorithms are intrinsically unconstrained optimization techniques. Constraint handling is mostly incorporated additionally and its choice has great bearing on the quality of the solution. Stochastic Ranking was introduced as an improvement over Feasibility Rules for handling constraints in Evolutionary Optimization. It is widely believed that stochastic ranking is currently the best-known technique for handling constraints. However, a fair comparative study has never been attempted in the literature, where by the performance of both the constraint handling technique is compared on the same Evolutionary Algorithm. This paper fairly compares the performance of both the constraint handling techniques on the same Evolutionary Algorithm over a set of parameters like feasibility rate, successful run, success rate and success performance in addition to objective function value and number of function evaluations. The results put a question mark over the belief that Feasibility Rules are worse than Stochastic Ranking.

Aimed at improving the insufficient search ability of constraint differential evolution with single constraint handling technique when solving complex optimization problem, this paper proposes a constraint differential evolution algorithm based on ensemble of constraint handling techniques and multi-population framework, called ECMPDE. First, handling three improved variants of differential evolution algorithms are dynamically matched with two constraint handling techniques through the constraint allocation mechanism. Each combination includes three variants with corresponding constraint handling technique and these combinations are in the set. Second, the population is divided into three smaller subpopulations and one larger reward subpopulation. Then a combination with three constraint algorithms is randomly selected from the set, and the three constraint algorithms are run in three sub-populations respectively. According to the improvement of fitness value, the optimal constraint algorithm is selected to run on the reward sub-population, which can share information and close cooperation among populations. In order to verify the effectiveness of the proposed algorithm, 12 standard constraint optimization problems and 10 engineering constraint optimization problems are tested. The experimental results show that ECMPDE is an effective algorithm for solving constraint optimization problems.

Several constraint handling techniques have been proposed to be used with the evolutionary algorithms (EAs). According to the no free lunch theorem, it is impossible for a single constraint handling technique to outperform all other techniques on every problem. In other words, depending on several factors such as the ratio between feasible search space and the whole search space, multi-modality of the problem, the chosen EA and global exploration/local exploitation stages of the search process, different constraint handling techniques can be effective on different problems and during different stages of the search process. Motivated by these observations, we proposed an ensemble of constraint handling techniques (ECHT) to solve constrained real-parameter optimization problems. In ECHT, each constraint handling method has its own population and every function call is used effectively. Being a general concept, the ECHT can be realized with any existing EA. In this paper, we present ECHT with Differential Evolution (DE) as the basic search algorithm (ECHT-DE). The ECHT is formed using four different constraint handling techniques present in the literature. ECHT-DE is evaluated on the functions from CEC 2010 problem set.

How to balance diversity and convergence as well as constraint violation and objective function in Constrained Optimization Problems (COPs) is an important and key issue when designing a new Constrained Optimization Evolutionary Algorithms (COEAs). C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> oDE set a good example on this point, and obtains promising results. In C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> oDE, the two Constraint Handling Techniques (CHTs) are adopted fixed in two phases. To verify whether there are some other more efficient combinations, Stochastic Ranking (SR) is adopted to balance the combination. Experimental results show a good performance, which also give some inspiration for other research.

Economic Load Dispatch (ELD) optimization is an important and difficult task in power system planning. Previously, most of the research mainly focused on proposing various evolutionary algorithms (EAs) to pursue better results of ELD problems. However, few comprehensive analysis of the effects of various constraint handling techniques (CHTs) on the performance of EA-based techniques are available so far. In this paper, we try to fill this gap by experimentally testing the algorithmic variants of combining four effective and widely used EAs with three CHTs. From the experimental results on the ELD problems with valve-point and those problems with both valve-point and multiple-fuel effect, several important conclusions can be achieved, including 1) for the low scale ELD problem with valve-point only, the selection of EAs is more important than CHTs; 2) for the large scale problems, CHTs play crucial roles; 3) the appropriate combination of EA and CHT is helpful to achieve better performance. This study is also expected to provide solid basis for further strengthening the robustness of EAs for ELD optimization. It is also interesting to observe that the experimental results obtained in this paper are much better than those of the previous effective ELD optimization algorithms.

In recent years, Differential Evolution (DE) algorithm that is one of the most important delegate of the evolutionary algorithms have been successfully used for solving different optimization problems with its both serial and parallel variants. In this study, the performance of DE algorithm powered with cooperative emigrant creation strategy was investigated over the different types of numerical problems and obtained results were compared with the results of the serial DE and parallel DE algorithm supported with conventional emigrant creation strategy. Experimental studies showed that the convergence speed of the parallel DE algorithm is significantly improved with the cooperative emigrant creation strategy and the proposed parallel DE algorithm produces better results compared to the serial DE algorithm and parallel DE algorithm with the conventional emigrant creation strategy.

A rank-based constraint handling technique for engineering design optimization problems solved by genetic algorithms

This paper proposes an application of Evolutionary Algorithms (EAs) such as Differential Evolution (DE), Evolutionary Programming (EP), Real coded Genetic Algorithm (RGA), Covariance Matrix Adaptation Evolution Strategy (CMAES) and Particle Swarm Optimization (PSO) to Reactive Power Planning (RPP) problem. RPP is a non-smooth and non-differentiable optimization problem for a multi-objective function. Three loading conditions (Normal load, 1.25% and 1.5% of normal load) have been considered. The IEEE 30 bus system is used to validate the effectiveness of the Evolutionary Algorithms. Simulation results shows that, the DE algorithm gives better results compared to other algorithms for all the loading conditions and it can be better suited for the multi objective RPP problem.

Differential evolution (DE) algorithm is a global optimization algorithm over continuous search space. DE also has been applied in many fields, such as artificial neural networks, chemical engineering, mechanical design, robotics, signal processing, biological information, and economics. At the same time, as a powerful evolutionary algorithm for solving global numerical optimization problems, the DE algorithm has drawn more and more attention. However, how to make a proper balance between the global and local search is a burning question and to limit the optimization performance of DE. In this paper, an improved algorithm η_CODE with a new η_Cauchy operator is proposed to enhance the global and local search ability of a well-known DE variant JADE. In order to guarantee the effective performance of the proposed operator, all the fitness values are ranked through a ranking scheme based on increasing order before a new η_Cauchy operator is conducted. The pNP individuals that have better fitness are selected and carried out Cauchy disturbance operation considering the complexity of the algorithm. The Dynamic parameter mechanism is utilized to select pNP individuals that number is also adjusted dynamically in each generation. The scale factor F and crossover probability CR are obtained with Lehmer mean without using determined parameter c in JADE, which aims to balance the exploration and exploitation of the algorithm during the running time. A total of sixty benchmark functions from CEC2014 and CEC2017 on real parameter optimization are employed to prove the validity of η_CODE for solving complex high-dimensional problems. The experiments indicate that η_CODE is better than or at least comparable with several state-of-the-art DE variants, including JADE, SinDE, TSDE, AGDE, and EFADE in the global numerical optimization problems. In order to further analyze the performance of η_CODE, we also select extra two high-powered modified algorithms called EBLSHADE and LSHADESPACMA based on LSHADE to discuss advantages and disadvantages of the proposed algorithm.

A differential evolution algorithm with intersect mutation operator

Application of a Derivative-Free Method with Projection Skill to Solve an Optimization Problem

In this paper we consider a cognitive radio network consisting of primary users (PUs) and a set of secondary users (SUs). The spectrum has been divided into channels using frequency division multiple access (FDMA). The PUs are licensed to use the channels. When the channels are idle, i.e., not used by the PUs then PUs lease the vacant spectrum for earning revenue. While the SUs bid for the channels the PUs selects the purchaser offering the highest of all bid values. The main objective is to benefit the seller. Here, using Real Coded Genetic Algorithm (RCGA) we solve the above mentioned single objective function. We also optimize the problem using differential evolution (DE) algorithm and make a comparison of fitness values with RCGA.

The Differential Evolution (DE) algorithm, under the family of Evolutionary Algorithms (EAs), is one of the powerful algorithms used for solving continuous parameter optimization challenges. The simplistic nature and robustness of the classical DE algorithm have drawn researchers' attention towards its progressive enhancement. This work reports on an investigation of the behavioral changes of the classical DE algorithm, evoked when its mutation and crossover components are fine tuned for enhancement of DE's performance. The scope of this study covers the implementation of a mutation level enhancement and a crossover level enhancement, followed by their integration. The mutation and the crossover components are augmented by incorporation of Centroid DE and Superior-Superior & Superior-Inferior DE logics, respectively. The algorithms appraised in this inquiry were classical DE, Centroid based DE(cDE), Superior-Superior based DE (ssDE), Superior-Inferior DE (siDE), Centroid Superior-Superior DE (cssDE) and Centroid Superior-Inferior DE (csiDE). These algorithms were evaluated by comparison of the values of their mean objective function (MOV), and their speed, at solving the global optimization problems in a simple benchmarking function suite with 4 functions of different categories. The study concludes that the DE algorithm shows enhancement performance with modified mutation and crossover components. However, with a trend for inconsistency for varying values of its control parameters and benchmarking problems.

In this work, a method to control the parameters of Differential Evolution (DE) algorithm is proposed. Here the control parameters of DE are co-evolved by Particle Swarm Optimization (PSO) algorithm. The classical DE algorithm has two main control parameters: Scale Factor (F) and Cross-over Rate (CR). These are selected on trial-and-error basis for solving optimization problems. Several optimization problems lead to optimal or sub-optimal solution by proper selection of control parameters of the DE algorithm. In this proposed method, PSO algorithm is used to tune the scale factor and cross-over rate in DE algorithm. Basically PSO algorithm is used as a meta-optimizer for DE algorithm. The proposed method is termed as mPSO-DE in this paper. The mPSO-DE algorithm is applied on 12 benchmark unconstrained optimization problems. The obtained results are compared with that of classical DE algorithm. From the experimental studies, it has been found that the proposed mPSO-DE algorithm performed better than DE algorithm.

Based on differential evolution algorithm and the concept of multi-objective optimization, a differential evolution algorithm for constrained optimization problem based on multi-objective constraint handling is proposed in this paper. This proposed algorithm can achieve global search and local search effectively by using of the differential evolution algorithm. For the constrained optimization problems, this paper presents a new comparison mechanism based on the concept of Pareto optimal solution. The grades of Pareto optimal solutions, feasible solutions and infeasible solutions are prescribed and enable them to be selected at different probabilities during the evolutionary process, which can lead the search towards the direction of the global optimum. In addition, an infeasible solution of replace mechanism is also given. When the algorithm gets into a local optimal, we use infeasible solution, which contains useful information, to replace those redundant and repeat feasible solutions to improve the exploration capabilities of the algorithm in the search space. Compared to Evolutionary Algorithm based on Homomorphous Maps (EAHM), Artificial Immune Response Constrained Evolutionary Strategy (AIRCES), Constraint Handling Differential Evolution (CHDE) and Evolutionary Strategies based on Stochastic Ranking (ESSR), the results of the 13 Standard tests show that the proposed algorithm has certain advantages in the convergence speed and solution accuracy.