Constrained Benchmark Functions Research Articles

Dynamic network structured immune particle swarm optimisation with small-world topology

Particle swarm optimisation (PSO) has attracted much attention and is used to wide applications in different fields in recent years because of its simple concept, easy implementation and quick convergence. However, it suffers from premature convergence since the population's diversity loses quickly. In this paper, a novel and efficient variant of PSO named DNIPSO is proposed which help the diversity of the swarm be preserved via the Newman-Watts small world network topology and the immune learning operator. Initially the topology of the population is the regular network. Then the Newman-Watts small world topology is formed gradually and the swarm evolves simultaneously. The optimisation process contains the population structure dynamics and particle immune learning two parts which mutually promoted effectively in whole population. Furthermore, the immune operator which is based on the clonal selection theory achieves a trade-off between exploration and exploitation abilities. Numerical experiments both on continuous unconstrained and constrained benchmark functions are used to test the performance of DNIPSO. Simulation results show it is effective and robust.

Enhanced grey wolf optimisation algorithm for constrained optimisation problems

Grey wolf optimiser (GWO) is a recent, fast and easy-to-implement, nature inspired meta-heuristic optimisation algorithm that focuses on social behaviour of grey wolves. GWO algorithm is prominent in terms of finding global optima without getting trapped in premature convergence. In order to find a fast convergent behaviour of GWO, an enhanced grey wolf optimisation (EGWO) algorithm is proposed in this paper. Basically, GWO is modified in two ways in this study, first, to improve exploitation capability of GWO, the hunting mechanism makes the best use of the global best solution, i.e., alpha and secondly, a random parameter of existing GWO algorithm is emended in order to produce promising results compared to state-of-the-art algorithms. To validate the effectiveness of proposed EGWO algorithm, penalty function is consolidated and diverse experiments are executed on different constrained benchmark functions of different complexities and characteristics. Further, a classical engineering design problem (pressure vessel) is solved using the proposed algorithm. The performance evaluation of proposed EGWO algorithm along with other standard meta-heuristic optimisation algorithms proved that the proposed EGWO algorithm to be a competitive algorithm in the field of nature inspired meta-heuristic optimisation algorithms.

Enhanced grey wolf optimisation algorithm for constrained optimisation problems

Grey wolf optimiser (GWO) is a recent, fast and easy-to-implement, nature inspired meta-heuristic optimisation algorithm that focuses on social behaviour of grey wolves. GWO algorithm is prominent in terms of finding global optima without getting trapped in premature convergence. In order to find a fast convergent behaviour of GWO, an enhanced grey wolf optimisation (EGWO) algorithm is proposed in this paper. Basically, GWO is modified in two ways in this study, first, to improve exploitation capability of GWO, the hunting mechanism makes the best use of the global best solution, i.e., alpha and secondly, a random parameter of existing GWO algorithm is emended in order to produce promising results compared to state-of-the-art algorithms. To validate the effectiveness of proposed EGWO algorithm, penalty function is consolidated and diverse experiments are executed on different constrained benchmark functions of different complexities and characteristics. Further, a classical engineering design problem (pressure vessel) is solved using the proposed algorithm. The performance evaluation of proposed EGWO algorithm along with other standard meta-heuristic optimisation algorithms proved that the proposed EGWO algorithm to be a competitive algorithm in the field of nature inspired meta-heuristic optimisation algorithms.

Optimization with a novel hybrid algorithm and applications

A novel hybrid algorithm (DPD) by combination of Differential Evolution (DE) and Particle Swarm Optimization (PSO) is proposed for solving unconstrained and constrained optimization problems. It is based on ‘tri-population’ environment. Initially, the whole population (in increasing order of fitness) is divided into three groups—Inferior Group, Mid Group and Superior Group. DE is employed in the inferior and superior groups, whereas PSO is used in the mid-group. Two strategies namely Elitism (to retain the best obtained values so far) and Non-redundant search (to maintain diversity in the population) have been employed in DPD cycle. Since the performance of DE is sensitive to the choice of the mutation strategy. Therefore suitable mutation strategy for both DEs used in DPD is investigated over a set of 8 popular mutation strategies. Combination of 8 mutation strategies is generated 64 different variants of DPD. Top 4 DPDs are investigated through 20 benchmark function. Based on the ‘performance’ analysis best DPD is reported using 20 unconstrained benchmark functions, 13 constrained benchmark functions, 6 unconstrained and 3 constrained engineering design problems. To show superiority and effectiveness, best DPD is compared with various state-of-the-art approaches in terms of quality of solutions.

Parallel hybridization of differential evolution and particle swarm optimization for constrained optimization with its application

This paper presents a novel hybridization between differential evolution (DE) and particle swarm optimization (PSO), based on ‘tri-population’ environment. Initially, the whole population (in increasing order of fitness) is divided into three groups—inferior group, mid group and superior group. DE is employed in the inferior and superior groups, whereas PSO is used in the mid-group. This proposed method is named as DPD as it uses DE–PSO–DE on the sub-populations of the same population. Two more strategies namely Elitism (to retain the best obtained values so far) and Non-Redundant Search (to improve the solution quality) have been incorporated in DPD cycle. Considering eight variants of popular mutation operators in one DE, a total of 64 variants of DPD are formed. The top four DPDs have been pointed out through 13 constrained benchmark functions and five engineering design problems. Further, based on the ‘performance’ analysis the best DPD is reported. Later to show superiority and effectiveness, the best DPD is compared with various state-of-the-art approaches. The numerical, statistical and graphical analyses reveal the robustness of the proposed DPD.

A Novel DE-PSO-DE (DPD) Algorithm for Economic Load Dispatch Problem

This paper presents a hybrid algorithm of two popular heuristics namely Differential Evolution (DE) and Particle Swarm Optimization (PSO) on a tri-population environment. Initially, the whole population (in increasing order of fitness) is divided into three groups – Inferior Group, Mid Group and Superior Group. DE is employed in the inferior and superior groups, whereas PSO is used in the mid-group. It is based on the information sharing mechanism of their inherent property to overcome the shortcomings of each other. The proposed method is called DPD as it uses DE-PSO-DE on a population. Two strategies namely Elitism (to retain the best obtained values so far) and Non-redundant search (to improve the solution quality) have been employed in DPD cycle. Out of a total of 64 DPDs, Top 4 DPDs are investigated through CEC2006 constrained benchmark functions. Based on the ‘performance' analysis, best DPD is reported and further used in solving 5 engineering design problems along with economic load dispatch problem in order to confirm further the efficiency of the proposed DPD.

Stochastic Fractal Search: A powerful metaheuristic algorithm

Evolutionary Algorithms (EAs) are well-known terms in many science fields. EAs usually interfere with science problems when common mathematical methods are unable to provide a good solution or finding the exact solution requires an unreasonable amount of time. Nowadays, many EA methods have been proposed and developed. Most of them imitate natural behavior, such as swarm animal movement. In this paper, inspired by the natural phenomenon of growth, a new metaheuristic algorithm is presented that uses a mathematic concept called the fractal. Using the diffusion property which is seen regularly in random fractals, the particles in the new algorithm explore the search space more efficiently. To verify the performance of our approach, both the constrained and unconstrained standard benchmark functions are employed. Some classic functions including unimodal and multimodal functions, as well as some modern hard functions, are employed as unconstrained benchmark functions; On the other hand, some well-known engineering design optimization problems commonly used in the literature are considered as constrained benchmark functions. Numerical results and comparisons with other state of the art stochastic algorithms are also provided. Considering both convergence and accuracy simultaneously, experimental results prove that the proposed method performs significantly better than other previous well-known metaheuristic algorithms in terms of avoiding getting stuck in local minimums, and finding the global minimum.

A novel artificial bee colony algorithm with Powell's method

Artificial bee colony (ABC) algorithm is a relatively new optimization technique which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in ABC regarding its solution search equation, which is good at exploration but poor at exploitation. To address this concerning issue, we first propose a modified search equation which is applied to generate a candidate solution in the onlookers phase to improve the search ability of ABC. Further, we use the Powell's method as a local search tool to enhance the exploitation of the algorithm. The new algorithm is tested on 22 unconstrained benchmark functions and 13 constrained benchmark functions, and are compared with some other ABCs and several state-of-the-art algorithms. The comparisons show that the proposed algorithm offers the highest solution quality, fastest global convergence, and strongest robustness among all the contenders on almost all test functions.

Teaching–learning-based optimization algorithm for unconstrained and constrained real-parameter optimization problems

An efficient optimization algorithm called teaching–learning-based optimization (TLBO) is proposed in this article to solve continuous unconstrained and constrained optimization problems. The proposed method is based on the effect of the influence of a teacher on the output of learners in a class. The basic philosophy of the method is explained in detail. The algorithm is tested on 25 different unconstrained benchmark functions and 35 constrained benchmark functions with different characteristics. For the constrained benchmark functions, TLBO is tested with different constraint handling techniques such as superiority of feasible solutions, self-adaptive penalty, ϵ-constraint, stochastic ranking and ensemble of constraints. The performance of the TLBO algorithm is compared with that of other optimization algorithms and the results show the better performance of the proposed algorithm.

A note on teaching–learning-based optimization algorithm

Teaching–Learning-Based Optimization (TLBO) seems to be a rising star from amongst a number of metaheuristics with relatively competitive performances. It is reported that it outperforms some of the well-known metaheuristics regarding constrained benchmark functions, constrained mechanical design, and continuous non-linear numerical optimization problems. Such a breakthrough has steered us towards investigating the secrets of TLBO’s dominance. This paper reports our findings on TLBO qualitatively and quantitatively through code-reviews and experiments, respectively. Our findings have revealed three important mistakes regarding TLBO: (1) at least one unreported but important step; (2) incorrect formulae on a number of fitness function evaluations; and (3) misconceptions about parameter-less control. Additionally, unfair experimental settings/conditions were used to conduct experimental comparisons (e.g., different stopping criteria). The experimental results for constrained and unconstrained benchmark functions under fairly equal conditions failed to validate its performance supremacy. The ultimate goal of this paper is to provide reminders for metaheuristics’ researchers and practitioners in order to avoid similar mistakes regarding both the qualitative and quantitative aspects, and to allow fair comparisons of the TLBO algorithm to be made with other metaheuristic algorithms.

Opposition‐Based Barebones Particle Swarm for Constrained Nonlinear Optimization Problems

This paper presents a modified barebones particle swarm optimization (OBPSO) to solve constrained nonlinear optimization problems. The proposed approach OBPSO combines barebones particle swarm optimization (BPSO) and opposition‐based learning (OBL) to improve the quality of solutions. A novel boundary search strategy is used to approach the boundary between the feasible and infeasible search region. Moreover, an adaptive penalty method is employed to handle constraints. To verify the performance of OBPSO, a set of well‐known constrained benchmark functions is used in the experiments. Simulation results show that our approach achieves a promising performance.

An effective differential evolution with level comparison for constrained engineering design

Solving constrained engineering design problems via evolutionary algorithms has attracted increasing attention in the past decade. In this paper, a simple but effective differential evolution with level comparison (DELC) is proposed for constrained engineering design problems by applying the level comparison to convert the constrained optimization problem into an unconstrained one and using the differential evolution (DE) to perform a global search over the solution space. In addition, the mutation factor of DE is set to be a random number to enrich the search behavior, and the satisfaction level increases monotonously to gradually stress the feasibility. The comparison results between the DELC and five existing algorithms from the literature based on 13 widely used constrained benchmark functions show that the DELC is of better or competitive performance. Furthermore, the DELC is used to solve some typical engineering design problems. DELC is of superior searching quality on all the problems with fewer evaluation times than other algorithms. In addition, the effect of the increasing rate of satisfaction level on the performances of the DELC is investigated as well.

Quantum-Inspired Immune Clonal Algorithm for Global Optimization

Based on the concepts and principles of quantum computing, a novel immune clonal algorithm, called a quantum-inspired immune clonal algorithm (QICA), is proposed to deal with the problem of global optimization. In QICA, the antibody is proliferated and divided into a set of subpopulation groups. The antibodies in a subpopulation group are represented by multistate gene quantum bits. In the antibody's updating, the general quantum rotation gate strategy and the dynamic adjusting angle mechanism are applied to accelerate convergence. The quantum not gate is used to realize quantum mutation to avoid premature convergences. The proposed quantum recombination realizes the information communication between subpopulation groups to improve the search efficiency. Theoretical analysis proves that QICA converges to the global optimum. In the first part of the experiments, 10 unconstrained and 13 constrained benchmark functions are used to test the performance of QICA. The results show that QICA performs much better than the other improved genetic algorithms in terms of the quality of solution and computational cost. In the second part of the experiments, QICA is applied to a practical problem (i.e., multiuser detection in direct-sequence code-division multiple-access systems) with a satisfying result.

Using quantum-behaved particle swarm optimization algorithm to solve non-linear programming problems

In this paper, we focus on solving non-linear programming (NLP) problems using quantum-behaved particle swarm optimization (QPSO). After a brief introduction to the original particle swarm optimization (PSO), we describe the origin and development of QPSO, and the penalty function method for constrained NLP problems. The performance of QPSO is tested on some unconstrained and constrained benchmark functions and compared with PSO with inertia weight (PSO-In) and PSO with constriction factor (PSO-Co). The experimental results show that QPSO outperforms the traditional PSOs and is a promising optimization algorithm.