An Improved C2oDE based on Stochastic Ranking

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.

Constrained optimization problems are common in theoretical study and engineering practice. The constraint functions can be equality or inequality, convex or non-convex, which makes it difficult to find a constrained optimal solution. Evolutionary algorithms are powerful tools for constrained optimization problems. The balance between constraints and objective functions is always a difficult task and hotspot in EA-community. Many EA-based constraint handling methods have been proposed, but this problem is far to be completely solved.This paper presents a new view of constraint handling: A bi-level differential evolutionary algorithm (BLDE). BLDE has a bi-level structure, which deals with the constraints and searches optimal solutions separately. In the lower level, the objective is to find a population of candidate feasible solutions, and in the upper level, the objective is to search the global optimal solution. The proposed BLDE gets competitive results in 13 benchmark functions, with a comparison of two state-of-the-art constraint handling algorithms. BLDE shows the following advantages: 1) it is efficient in search constrained optimal solutions; 2) it is easy to implement and requires few hyper-parameters; 3) it can guarantee the obtained solutions are feasible or close to feasible.

A fast constrained state transition algorithm

An empirical comparison is conducted to analyze three different elements involved in population diversity management in constrained numerical optimization problems: (1) optimization algorithms, (2) constraint-handling techniques and (3) diversity promotion techniques. Combinations of basic algorithm versions, differential evolution, and real-coded genetic algorithm, applying feasibility rules, $\epsilon$ -Constrained method and stochastic ranking as constraint-handling techniques, and also implementing a diversity promotion technique, were generated. Each combination (48 in total) was tested by solving 36 well-known benchmark problems. The final results highlight the combination of differential evolution, $\epsilon$ -Constrained method and a couple of non-disruptive diversity promotion techniques, as the most promising combinations.

Real world problems are often formularised as constrained optimisation problems (COPs). Constraint handling techniques are important for efficient search, and various approaches such as penalty methods or feasibility rule have been studied. The penalty methods deal with a single fitness function by combining an objective function value and a constraint violation with a penalty factor. Moreover, the penalty factor can be flexibly adapted by feeding back information on search process in adaptive penalty methods. However, setting parameters properly and keeping the good balance between the objective function value and the constraint violation are difficult. In this paper, we propose a new parameter-free adaptive penalty method with balancing the objective function value and the constraint violation. L-SHADE is adopted as a base search algorithm, and the optimisation results of 28 benchmark functions provided by the CEC2017 and CEC2018 competitions on constrained single-objective numerical optimisations are compared with other methods.

Generally, constraint-handling techniques are designed for evolutionary algorithms to solve Constrained Multiobjective Optimization Problems (CMOPs). Most Multiojective Particle Swarm Optimization (MOPSO) designs adopt these existing constraint-handling techniques to deal with CMOPs. In this chapter, the authors present a constrained MOPSO in which the information related to particles' infeasibility and feasibility status is utilized effectively to guide the particles to search for feasible solutions and to improve the quality of the optimal solution found. The updating of personal best archive is based on the particles' Pareto ranks and their constraint violations. The infeasible global best archive is adopted to store infeasible nondominated solutions. The acceleration constants are adjusted depending on the personal bests' and selected global bests' infeasibility and feasibility statuses. The personal bests' feasibility statuses are integrated to estimate the mutation rate in the mutation procedure. The simulation results indicate that the proposed constrained MOPSO is highly competitive in solving selected benchmark problems.

The relationship between Constraint Handling Techniques (CHTs) and Constrained Optimization Problems (COPs) is an important issue as it gives some direction on deciding when and how to use these CHTs effectively. In the previous studies, the relationship between problems and algorithms as well as the CHTs are studied from the qualitative and quantitative aspects. This paper tries to study the relationship between COPs and two simple CHTs, i.e., Deb’s feasibility-based rule and multiobjective optimization technique, with a new set of benchmark functions adopted. The experimental results show the importance of the classification of problems, which is a basis for related research.

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.

In this paper a hybrid adaptive genetic algorithm is proposed for solving constrained optimization problems. Genetic algorithm proposed here combines adaptive penalty method and smoothing technique in order to make the algorithm not needing parameters tuning and easily escaping from the local optimal solutions. Meanwhile, local line search technique is introduced and a new crossover operator is designed for getting much faster algorithm convergence. If there is no feasible solutions in the current population, finding feasible solutions is prior to finding optimal solution, otherwise the exploitation for global optimal solution based on a certain smoothing function at the best feasible solution in the current population and the exploration for whole search space are processing at the same time. The performance of the algorithm is tested on thirteen benchmark functions in the literature and the results indicate that the algorithm proposed here is robust and effective.

Any evolutionary algorithm tends to end up in a local optimum. A new approach based on an evolutionary algorithm named as Diversity Driven Multi-Parent Evolutionary Algorithm with Adaptive non-uniform mutation is presented. In the proposed algorithm, Non-uniform mutation is used to maintain diversity in the explored solutions. Fitness variance, which signifies solution space aggregation, is used to detect the premature convergence of the population to a local optimum. The term multi-parent is used in the context of more than two parents participating in crossover operation. After multi-parent selection for cross-over to generate new solutions, the non-uniform adaptive mutation is performed, which in turn is triggered by the diminishing value of fitness variance of candidate solutions and pushes solutions out of local optimum. Hence, it can be said that the algorithm is driven by the diversity of the population and overcomes the tendency of evolutionary algorithms to stuck in local optimum. The performance of this algorithm is tested on 23 basic benchmarks, CEC05 functions, and CEC17 functions. As CEC17 benchmark functions include constraint problems, a constraint-handling technique is proposed based on the fuzzy set theory. In the proposed constrained handling strategy, constraint violation is also taken as another objective along with the main objective. The decision to accept or discard the solution is based on the fuzzy set theory. The values of constraint violation and objective function are calculated and fuzzified by calculating membership values by considering the main objective and constraint violation as triangular fuzzy functions. The best solutions are selected based on cardinal priority ranking. The obtained results from the proposed algorithm are compared with the results available in the literature. The result indicates that this algorithm is competitive, even with a smaller number of function evaluations.

This paper presents a novel boundary approach which is included as a constraint-handling technique in an ant colony algorithm. 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. Our proposed technique precisely focuses the search on the boundary region and can be either used alone or in combination with other constraint-handling techniques depending on the type and number of problem constraints. For validation purposes, an ant algorithm is adopted as our search engine. We compare our proposed approach with respect to constraint-handling techniques that are representative of the state-of-the-art in constrained evolutionary optimization using a set of standard test functions.

In this paper, a concurrent structure-control design is applied to a five-bar parallel robot in order to minimize the tracking error in a high-speed task. The high power necessary to perform the task leads to a challenging problem in handling a physical constraint on the maximum torque as well as on the structure-control design parameters. Three constraint-handling techniques, the feasibility rules, ε-constrained method and stochastic ranking have been used to analyze how the design changes as the torque constraint are reduced. Studying torque reduction is important from the energy consumption point of view, as well as the use of lighter and cheaper motors. In addition, a comparison among the three constraint-handling techniques is implemented so as to observe the performance and quality of the obtained results. The final results suggest that stochastic ranking method always obtains feasible and successful solutions in all proposed torque limits.

In this paper, a convex filled function method for solving constrained global optimization problems is proposed. A convex filled function is proposed by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a convex filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems.

An evolutionary algorithm with directed weights for constrained multi-objective optimization

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.