Efficient Methods for Constraint-Handling in Evolutionary Algorithm-Based Management of Irrigation Canal Network Operations

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.

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.

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.

Most of real-world optimization problems correspond to constrained multi-objective optimization problems (CMOPs). Multi-objective evolutionary algorithms (MOEAs) are useful to find out diverse Pareto-optimal solutions in CMOPs, and has been applied in various engineering fields. In order to expand the range of application of MOEAs further, we propose an improved evolutionary algorithm with parallel evaluation strategy (EAPES). In EAPES, feasible solutions and infeasible solution are separately stored in different populations, and infeasible solutions are evaluated in an unusual manner where not only feasible solutions but also useful infeasible solutions are used as parents to reproduce the populations for the next generation. Infeasible solutions are ranked based on a scalarizing function determined by objective function values and a total constraint violation value. This paper investigates the performance of the proposed EAPES to search for Pareto-optimal solutions compared to NSGA-II and the previous EAPES. The proposed EAPES with a well-tuned parameter is most capable to explore Pareto-optimal solutions with good diversity, spread, and convergence to the true Pareto front. The proposed EAPES assigns bad rank to the infeasible solutions that exist away from the true Pareto front, and does not store such solutions. Thus the proposed EAPES exhibits a higher searching capability than the previous EAPES by evaluating infeasible solutions in an appropriate balance between objective functions and total constraint violation. In addition, the results suggest that the proposed EAPES may exhibit high solution search performance even in more difficult CMOPs, which have more objective functions and/or more constraints.

Knapsack problems are among the most common problems in literature tackled with evolutionary algorithms (EA). Their major advantage lies in the fact that they are relatively simple to implement while they allow generalizations for a wide range of real world problems. The multi-dimensional knapsack problem (MKP), which belongs to the class of NP-complete combinatorial optimization problems, is one of the variations of the knapsack problem. The MKP has a wide range of real world applications such as cargo loading, selecting projects to fund, budget management, cutting stock, etc. The MKP has been studied quite extensively in the EA community. Due to the constrained nature of the problem, constraint handling techniques gain great importance in the performance of the proposed EA approaches. In this study, the applicability of a generational EA that uses a penalty-based constraint handling technique and a gene locus based, asymmetric, adaptive mutation scheme is explored for the MKP. The effects of the parameters of the explored approach is determined through tests. Further experiments, using large MKP instances from commonly used benchmarks available through the Internet are performed. Comparison tables are given for the performance of the explored approach and other good performing EAs found in literature for the MKP. Results show that performance improves greatly when compared with other penalty-based techniques, but the explored approach is still not the best performer among all. However, unlike the explored technique, the EAs using the other constraint handling techniques require a great amount of extra computational effort and need heuristic information specific to the optimization problem. Based on these observations, and the fact that the performance difference between the explored scheme and the better performers is not too high, research on improving the explored approach is still in progress.

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

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.

Almost all real-world and engineering problems involve multi-objective optimization of some sort that is often constrained. To solve these constrained multi-objective optimization problems, constrained multi-objective optimization evolutionary algorithms (CMOEAs) are enlisted. These CMOEAs require specific constraint handling techniques. This study aims to address and test the most successful constraint handling techniques, seven different penalty constraint techniques, as applied to the Non-dominated Sorting Genetic Algorithm II (NSGA-II). In this paper, NSGA-II is chosen because of its high popularity amongst evolutionary algorithms. Inverted Generational Distance and Hypervolume are the main metrics that are discussed to compare the constraint handling techniques. NSGA-II is applied on 13 constrained multi-objective problems known as CF1-CF10, C1-DTLZ1, C2-DTLZ2, and C3-DTLZ4. The result of IGD and HV values are compared and the feasibility proportions of each combination on each problem are shown. The results of simulation present interesting findings that have been presented at the end of paper as discussion and conclusion.

Highly constrained multiobjective optimization problems (HCMOPs) refer to constrained multiobjective optimization problems (CMOPs) with complex constraints and small feasible regions, which are commonly encountered in many real-world applications. Current constraint-handling techniques will face two difficulties when dealing with HCMOPs: 1) feasible solution is hard to be found and too much search effort is spent in locating the feasible region and 2) since the total feasible region of an HCMOP can consist of several disconnected subregions, the search process might be stuck in the comparatively larger feasible subregion, which does not contain the whole Pareto front (PF). To address these two issues, an evolutionary algorithm with constraint relaxation strategy based on differential evolution algorithm, that is, CRS-DE, is proposed in this article. In each generation, the CRS-DE relaxes the constraints by dividing the infeasible solutions into two subpopulations based on total constraint violation, that is, the ``semifeasible'' subpopulation (SF) and ``infeasible'' subpopulation (IF), respectively. The SF provides information on the promising regions of finding the feasible solution and is the driving force for convergence toward the PF, while the IF focuses on global exploration for new promising regions. Corresponding reproduction and selection strategies are devised for the SF, IF, and feasible subpopulations, which create a clear division of labor with cooperation to facilitate the search for feasible solutions. To leverage the influence of CRS and prevent the population from premature convergence, a mobility restriction mechanism is developed to restrict the individuals in the SF and IF from entering the feasible subpopulation and enhance the diversity of the whole population. Comprehensive experiments on a series of benchmark test problems and a real-world CMOP demonstrate the competitiveness of our method compared with other representative algorithms in terms of effectiveness and reliability in finding a set of well-distributed optimal solutions for HCMOPs.

In this paper, a special nonlinear bilevel programming problem (nonlinear BLPP) is transformed into an equivalent single objective nonlinear programming problem. To solve the equivalent problem effectively, we first construct a specific optimization problem with two objectives. By solving the specific problem, we can decrease the leader's objective value, identify the quality of any feasible solution from infeasible solutions and the quality of two feasible solutions for the equivalent single objective optimization problem, force the infeasible solutions moving toward the feasible region, and improve the feasible solutions gradually. We then propose a new constraint-handling scheme and a specific-design crossover operator. The new constraint-handling scheme can make the individuals satisfy all linear constraints exactly and the nonlinear constraints approximately. The crossover operator can generate high quality potential offspring. Based on the constraint-handling scheme and the crossover operator, we propose a new evolutionary algorithm and prove its global convergence. A distinguishing feature of the algorithm is that it can be used to handle nonlinear BLPPs with nondifferentiable leader's objective functions. Finally, simulations on 31 benchmark problems, 12 of which have nondifferentiable leader's objective functions, are made and the results demonstrate the effectiveness of the proposed algorithm.

Background: This article reviews recent patents pertaining to the technological achievements and shortcomings of aircraft Position Indicator and Actuator System (PIAS). Generally, most patents have concentrated more on design, while implementing various developed single or multiobjective algorithms to solve component or system optimal design problems such as those of PIAS. Thus, there is a noticeable lack of patent efforts on either developing new constraint-handling techniques especially for multi-objective optimization, or applying advance single-objective constrainthandling techniques to multi-objective optimization. Method: To address this, the authors of this article in there past work developed a multi-objective optimization algorithm called Genetic Pareto Set Identification Algorithm plus Different Sex (GPSIA+DS), which considers the feasible and infeasible solution in it optimization strategy. However, the application of GPSIA+DS to real world reliability engineering problem has not been validated. Result: Thus, in this work, GPSIA+DS ability to handle an aircraft PIAS optimization problem was presented. PIAS component failure rates were obtained via Fault Tree Analysis (FTA). PIAS cost and reliability were the study objective functions subject to weight constrains. The performance of GPSIA+DS for the test problem was investigated through a comparative study. Conclusion: In the comparative study GPSIA+DS was compared with fast-sort genetic algorithm plus constrained dominance principle (NSGA-II+CD). Although NSGA-II+DS out performed GPSIA+DS for this test problem, it is evident that GPSIA+DS is equally an efficient algorithm. Keywords: Aircraft system, constrain, failure rate, genetic algorithm, multi-objective optimization (MOO), pareto set, reliabilityAircraft system, constrain, failure rate, genetic algorithm, multi-objective optimization (MOO), pareto set, reliability

A feasible-ratio control technique for constrained optimization

Particle Swarm Optimization (PSO) is a brilliant swarm intelligence-based algorithm invented by Kennedy and Eberhart in 1995. Since its birth, it has been applied to many optimization problems to provide robust solutions in several engineering and applied sciences domains. The Cascaded Short Term Hydrothermal Scheduling (CSTHTS) problem, a highly non-linear, non-convex and multimodal scheduling problem, is solved by many metaheuristic algorithms, including PSO in the literature. However, It is found that the results reported on a standard benchmark test case of the CSTHTS problem are very poor compared to the results achieved on the same test case by implementing other advanced metaheuristic optimization algorithms. Therefore, It has been investigated and validated through the implementation that there can be an immense improvement in the performance of the PSO algorithm by just using another better constraint-handling approach. This article presents the improvement in the results of PSO on the CSTHTS problem as given in the literature by implementing a better constraint handling technique.

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.

In this paper, we present the constraint handling techniques developed to tackle a real world optimization application: the frequency assignment problem (FAP) in cellular radio networks. The main goal of this application consists in finding assignments which minimize electromagnetic interference due to frequency reuse and minimize the number of frequencies used. An acceptable assignment of the FAP must satisfy a set of multiple constraints such as traffic constraints and interference constraints. We present how each type of constraint can play a different role in EAs. In particular, we show how co-station constraints can be used to reduce the search space efficiently. The effectiveness of the constraint handling techniques combined with a regeneration technique is tested on big and hard FAP instances. Results show that limiting the degree of unfeasible solutions is beneficial for this application.