Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms in current use. DE operates through similar computational steps as employed by a standard evolutionary algorithm (EA). However, unlike traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used for generating the offspring. Since its inception in 1995, DE has drawn the attention of many researchers all over the world resulting in a lot of variants of the basic algorithm with improved performance. This paper presents a detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far. Also, it provides an overview of the significant engineering applications that have benefited from the powerful nature of DE.

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Differential Evolution: A Survey of the State-of-the-Art

Salp Swarm Algorithm

In this paper a novel nature-inspired optimization paradigm is proposed called Moth-Flame Optimization (MFO) algorithm. The main inspiration of this optimizer is the navigation method of moths in nature called transverse orientation. Moths fly in night by maintaining a fixed angle with respect to the moon, a very effective mechanism for travelling in a straight line for long distances. However, these fancy insects are trapped in a useless/deadly spiral path around artificial lights. This paper mathematically models this behaviour to perform optimization. The MFO algorithm is compared with other well-known nature-inspired algorithms on 29 benchmark and 7 real engineering problems. The statistical results on the benchmark functions show that this algorithm is able to provide very promising and competitive results. Additionally, the results of the real problems demonstrate the merits of this algorithm in solving challenging problems with constrained and unknown search spaces. The paper also considers the application of the proposed algorithm in the field of marine propeller design to further investigate its effectiveness in practice. Note that the source codes of the MFO algorithm are publicly available at http://www.alimirjalili.com/MFO.html.

Moth-flame optimization algorithm

This paper proposes a novel nature-inspired algorithm called Multi-Verse Optimizer (MVO). The main inspirations of this algorithm are based on three concepts in cosmology: white hole, black hole, and wormhole. The mathematical models of these three concepts are developed to perform exploration, exploitation, and local search, respectively. The MVO algorithm is first benchmarked on 19 challenging test problems. It is then applied to five real engineering problems to further confirm its performance. To validate the results, MVO is compared with four well-known algorithms: Grey Wolf Optimizer, Particle Swarm Optimization, Genetic Algorithm, and Gravitational Search Algorithm. The results prove that the proposed algorithm is able to provide very competitive results and outperforms the best algorithms in the literature on the majority of the test beds. The results of the real case studies also demonstrate the potential of MVO in solving real problems with unknown search spaces. Note that the source codes of the proposed MVO algorithm are publicly available at http://www.alimirjalili.com/MVO.html.

Multi-Verse Optimizer: a nature-inspired algorithm for global optimization

An adaptive particle swarm optimization (APSO) that features better search efficiency than classical particle swarm optimization (PSO) is presented. More importantly, it can perform a global search over the entire search space with faster convergence speed. The APSO consists of two main steps. First, by evaluating the population distribution and particle fitness, a real-time evolutionary state estimation procedure is performed to identify one of the following four defined evolutionary states, including exploration, exploitation, convergence, and jumping out in each generation. It enables the automatic control of inertia weight, acceleration coefficients, and other algorithmic parameters at run time to improve the search efficiency and convergence speed. Then, an elitist learning strategy is performed when the evolutionary state is classified as convergence state. The strategy will act on the globally best particle to jump out of the likely local optima. The APSO has comprehensively been evaluated on 12 unimodal and multimodal benchmark functions. The effects of parameter adaptation and elitist learning will be studied. Results show that APSO substantially enhances the performance of the PSO paradigm in terms of convergence speed, global optimality, solution accuracy, and algorithm reliability. As APSO introduces two new parameters to the PSO paradigm only, it does not introduce an additional design or implementation complexity.

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Adaptive Particle Swarm Optimization

Evolutionary algorithms are heuristic methods that have yielded promising results for solving nonlinear, nondifferentiable, and multi-modal optimization problems in the power systems area. The differential evolution (DE) algorithm is an evolutionary algorithm that uses a rather greedy and less stochastic approach to problem solving than do classical evolutionary algorithms, such as genetic algorithms, evolutionary programming, and evolution strategies. DE also incorporates an efficient way of self-adapting mutation using small populations. The potentialities of DE are its simple structure, easy use, convergence property, quality of solution, and robustness. This paper proposes a new approach for solving economic load dispatch problems with valve-point effect. The proposed method combines the DE algorithm with the generator of chaos sequences and sequential quadratic programming (SQP) technique to optimize the performance of economic dispatch problems. The DE with chaos sequences is the global optimizer, and the SQP is used to fine-tune the DE run in a sequential manner. The combined methodology and its variants are validated for two test systems consisting of 13 and 40 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. The proposed combined method outperforms other state-of-the-art algorithms in solving load dispatch problems with the valve-point effect.

Correction to "Combining of Chaotic Differential Evolution and Quadratic Programming for Economic Dispatch Optimization with Valve-Point Effect"

In this correspondence, an approach based on coevolutionary particle swarm optimization to solve constrained optimization problems formulated as min-max problems is presented. In standard or canonical particle swarm optimization (PSO), a uniform probability distribution is used to generate random numbers for the accelerating coefficients of the local and global s. We propose a Gaussian probability distribution to generate the accelerating coefficients of PSO. Two populations of PSO using Gaussian distribution are used on the optimization algorithm that is tested on a suite of well-known benchmark constrained optimization problems. Results have been compared with the canonical PSO (constriction factor) and with a coevolutionary genetic algorithm. Simulation results show the suitability of the proposed algorithm in terms of effectiveness and robustness

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Coevolutionary Particle Swarm Optimization Using Gaussian Distribution for Solving Constrained Optimization Problems

The identification of uncertain and nonlinear systems is an important and challenging problem. Fuzzy models, particularly Takagi-Sugeno (TS), have received particular attention in the area of nonlinear identification due to their potentialities to approximate any nonlinear behavior. A method of nonlinear identification based on the TS fuzzy model and optimization procedure is proposed in this paper. Chaotic particle swarm optimization (CPSO) algorithms, based on chaotic Zaslavskii map sequences, combined with efficient Gustafson-Kessel (GK) clustering algorithm are proposed here for the design of the premise part of production rules, while the least-mean-square technique is utilized for the subsequent part of the production rules of the TS fuzzy model. An experimental case study using a nonlinear yo-yo motion control system is analyzed by the proposed algorithms. The numerical results presented here indicate that the traditional particle swarm optimization algorithm and, particularly, the CPSO combined with GK algorithms are effective in building a good TS fuzzy model for nonlinear identification.

Fuzzy Identification Based on a Chaotic Particle Swarm Optimization Approach Applied to a Nonlinear Yo-yo Motion System

Optimization metaheuristics, such as Particle Swarm Optimization, Ant Colony Optimization and bacterial foraging strategies have become very popular in the optimization community and have been successfully applied to electromagnetic device design. The Artificial Bee Colony (ABC) algorithm is a rather new bio-inspired swarm intelligence approach which is competitive with other population-based algorithms and has the advantage of using fewer control parameters. In this work, a standard and an improved version of the ABC algorithm using Gaussian distribution are applied to Loney's solenoid problem, showing the suitability of these methods for electromagnetic optimization.

Gaussian artificial bee colony algorithm approach applied to Loney's solenoid benchmark problem

Studies with the Gaussian and Cauchy probability distributions have shown that the performance of the standard PSO algorithm can be improved. But these versions may also get stuck in local minima when optimizing functions with many local minima in high dimensional search space. In this paper, we will provide new results with PSO using the Exponential probability distribution aiming at improvement in performance. This version of the algorithm, termed PSO-E, was tested on a suite of well-known benchmark functions with many local optima and the results were compared with those obtained by the standard PSO (constriction factor). Simulation results show the suitability of PSO– E.

L. dos Santos Coelho

Papers

Correction to "Combining of Chaotic Differential Evolution and Quadratic Programming for Economic Dispatch Optimization with Valve-Point Effect"

Coevolutionary Particle Swarm Optimization Using Gaussian Distribution for Solving Constrained Optimization Problems

Fuzzy Identification Based on a Chaotic Particle Swarm Optimization Approach Applied to a Nonlinear Yo-yo Motion System

Gaussian artificial bee colony algorithm approach applied to Loney's solenoid benchmark problem

PSO-E: Particle Swarm with Exponential Distribution