An adaptive co-evolutionary competitive particle swarm optimizer for constrained multi-objective optimization problems

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

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.

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.

Push and pull search embedded in an M2M framework for solving constrained multi-objective optimization problems

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization.

Convergence, diversity and feasibility are three important issues when solving constrained multiobjective optimization problems (CMOPs). To deal with the balance among convergence, diversity and feasibility well, this article proposes a cooperative multiobjective evolutionary algorithm with propulsive population (CMOEA-PP) for solving CMOPs. CMOEA-PP has two populations, including propulsive population and normal population, and these two populations work cooperatively. Specifically, propulsive population focuses on convergence. Normal population gives priority to feasibility and is obligated to maintain diversity. To cross through the infeasible region and reach the Pareto front (PF), propulsive population does not consider constraints in the early stage and only considers constraints in the later stage. To further accelerate the speed of convergence, propulsive population only searches for corner solutions and center solutions, while normal population searches for the whole PF. As a result, propulsive population can cross through the infeasible region because of the lack of attention to feasibility. In addition, propulsive population also can guide and accelerate the convergence of the evolutionary process. Comprehensive experiment results on several sets of benchmark problems demonstrate that CMOEA-PP is better than existing state-of-the-art competitors.

Dual population approximate constrained Pareto front for constrained multiobjective optimization

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

This paper proposes an improved epsilon constraint-handling mechanism, and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA) is named MOEA/D-IEpsilon. It adjusts the epsilon level dynamically according to the ratio of feasible to total solutions (RFS) in the current population. In order to evaluate the performance of MOEA/D-IEpsilon, a new set of CMOPs with two and three objectives is designed, having large infeasible regions (relative to the feasible regions), and they are called LIR-CMOPs. Then the fourteen benchmarks, including LIR-CMOP1-14, are used to test MOEA/D-IEpsilon and four other decomposition-based CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D. The experimental results indicate that MOEA/D-IEpsilon is significantly better than the other four CMOEAs on all of the test instances, which shows that MOEA/D-IEpsilon is more suitable for solving CMOPs with large infeasible regions. Furthermore, a real-world problem, namely the robot gripper optimization problem, is used to test the five CMOEAs. The experimental results demonstrate that MOEA/D-IEpsilon also outperforms the other four CMOEAs on this problem.

A dual-population algorithm based on alternative evolution and degeneration for solving constrained multi-objective optimization problems

A feasible-ratio control technique for constrained optimization

Particle swarm optimization (PSO) algorithm has attracted great attention as a stochastic optimizing method due to its simplicity and power strength in optimization fields. However, two issues are still to be improved, especially, for complex multimodal problems. One is the premature convergence for multimodal problems. The other is the low efficiency for complex problems. To address these two issues, firstly, a strategy based on the global optimum prediction is proposed. A predicting model is established on the low-dimensional feature space with the principle component analysis technique, which has the ability to predict the global optimal position by the feature reflecting the evolution tendency of the current swarm. Then the predicted position is used as a guideline exemplar of the evolution process together with pbest and gbest. Secondly, a strategy, called adaptive mutation, is proposed, which can evaluate the crowding level of the aggregating particle swarm by using the distribution topology of each dimension, and hence, can get the possible location of local optimums and escape from the valleys with the generalized non-uniform mutation operator subsequently. The performance of the proposed global prediction-based adaptive mutation particle swarm optimization (GPAM-PSO) is tested on 8 well-known benchmark problems, compared with 9 existing PSO in terms of both accuracy and efficiency. The experimental results demonstrate that GPAM-PSO outperforms all reference PSO algorithms on both the solution quality and convergence speed.

The performance of the Particle Swarm Optimization (PSO) algorithm can be greatly improved if the parameters are appropriately tuned. However, tuning the control parameters for PSO algorithms has traditionally been a time-consuming, empirical process. Furthermore, ideal parameters may be time-dependent. To address the issue of parameter tuning, self-adaptive PSO (SAPSO) algorithms adapt the PSO control parameters over time. While many such SAPSO techniques have been proposed, their behaviour is not well understood as no in-depth critical analysis of their adaptation mechanisms has been performed. This study examines the convergence behaviour of eight SAPSO algorithms both analytically and empirically. Evidence clearly indicates that the field of self-adaptive PSO algorithms is in a sad state, given that many techniques either demonstrate divergent behaviour coupled with excessive invalid particles, and thus infeasible solutions, or have prohibitively low particle step sizes caused by rapid convergence.

Aiming at the control variables of reactive power optimization are discrete, and some parameters in the standard particle swarm optimization (PSO) algorithm need to be predefined by test, so the algorithm's practicability is restricted. For these reasons, an adaptive particle swarm optimization (APSO) algorithm is proposed by the authors. APSO introduces the self-adaptive tuning strategy and boundary constraint conditions can find the global optimal solution and solve the discrete variables. The reactive power optimization results of the standard IEEE-30-bus power system show that APSO is efficient than standard PSO. The global convergence accuracy and convergence stability is obviously improved compared with that of PSO.

Network reconfiguration for loss reduction in a distribution system is a very important way to save the electrical energy. This article proposes a hybrid evolutionary algorithm to solve the distribution feeder reconfiguration problem. The algorithm combines a fuzzy adaptive particle swarm optimization with a differential evolution algorithm, called the fuzzy adaptive particle swarm optimization–differential evolution. The fuzzy adaptive particle swarm optimization includes two parts. The first part is fuzzy adaptive binary particle swarm optimization, which determines the status of tie switches (open or closed), and the second part is fuzzy adaptive discrete particle swarm optimization, which determines the sectionalizing switch number. In the proposed algorithm, due to the results of binary particle swarm optimization and discrete particle swarm optimization algorithms that highly depend on the values of their parameters (such as the inertia weight and learning factors), a fuzzy system is employed to adaptively adjust the parameters during the search process. The differential evolution algorithm is combined with the fuzzy adaptive particle swarm optimization algorithm to improve its performance. The proposed algorithm is tested on two distribution test feeders. The results of simulation show that the proposed method is very powerful and is guaranteed to obtain the global optimization.