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

Having developed multiobjective optimization algorithms using evolutionary optimization methods and demonstrated their niche on various practical problems involving mostly two and three objectives, there is now a growing need for developing evolutionary multiobjective optimization (EMO) algorithms for handling many-objective (having four or more objectives) optimization problems. In this paper, we recognize a few recent efforts and discuss a number of viable directions for developing a potential EMO algorithm for solving many-objective optimization problems. Thereafter, we suggest a reference-point-based many-objective evolutionary algorithm following NSGA-II framework (we call it NSGA-III) that emphasizes population members that are nondominated, yet close to a set of supplied reference points. The proposed NSGA-III is applied to a number of many-objective test problems with three to 15 objectives and compared with two versions of a recently suggested EMO algorithm (MOEA/D). While each of the two MOEA/D methods works well on different classes of problems, the proposed NSGA-III is found to produce satisfactory results on all problems considered in this paper. This paper presents results on unconstrained problems, and the sequel paper considers constrained and other specialties in handling many-objective optimization problems.

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An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints

This paper proposes a novel population-based optimization algorithm called Sine Cosine Algorithm (SCA) for solving optimization problems. The SCA creates multiple initial random candidate solutions and requires them to fluctuate outwards or towards the best solution using a mathematical model based on sine and cosine functions. Several random and adaptive variables also are integrated to this algorithm to emphasize exploration and exploitation of the search space in different milestones of optimization. The performance of SCA is benchmarked in three test phases. Firstly, a set of well-known test cases including unimodal, multi-modal, and composite functions are employed to test exploration, exploitation, local optima avoidance, and convergence of SCA. Secondly, several performance metrics (search history, trajectory, average fitness of solutions, and the best solution during optimization) are used to qualitatively observe and confirm the performance of SCA on shifted two-dimensional test functions. Finally, the cross-section of an aircraft's wing is optimized by SCA as a real challenging case study to verify and demonstrate the performance of this algorithm in practice. The results of test functions and performance metrics prove that the algorithm proposed is able to explore different regions of a search space, avoid local optima, converge towards the global optimum, and exploit promising regions of a search space during optimization effectively. The SCA algorithm obtains a smooth shape for the airfoil with a very low drag, which demonstrates that this algorithm can highly be effective in solving real problems with constrained and unknown search spaces. Note that the source codes of the SCA algorithm are publicly available at http://www.alimirjalili.com/SCA.html .

SCA: A Sine Cosine Algorithm for solving optimization problems

Salp Swarm Algorithm

Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of multiobjective evolutionary algorithms for dealing with complicated PS shapes. It also proposes a new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE, and compares the proposed algorithm with NSGA-II with the same reproduction operators on the test instances introduced in this paper. The experimental results indicate that MOEA/D could significantly outperform NSGA-II on these test instances. It suggests that decomposition based multiobjective evolutionary algorithms are very promising in dealing with complicated PS shapes.

Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II

Solving sparse optimization problems via regularization frameworks is the dominant methodology for reconstructing sparse signals in the area of compressive sensing. In recent a few years, the use of multiobjective evolutionary algorithms (MOEAs) for sparse optimization has also attracted some research interests. Under the multiobjective framework, the loss term (error) and the regularization term (sparsity) are treated as two separate objective functions. So far, two popular multiobjective frameworks, NSGA-II and MOEA/D, have been used for sparse optimization. In this paper, we further develop a new MOEA/D variant for sparse reconstruction and sparsity detection, which involves three phases — approximating Pareto front (PF) in a chain order (phase 1) and in a random order (phase 2), and exploiting a knee region (phase 3 — optional). Our experimental results show that our proposed method is more effective than the earlier version of MOEA/D and the HALF solver in sparse signal reconstruction and sparsity detection.

A multi-phase multiobjective approach based on decomposition for sparse reconstruction

Recently, the research on quantum-inspired evolutionary algorithms (QEA) has attracted some attention in the area of evolutionary computation. QEA use a probabilistic representation, called Q-bit, to encode individuals in population. Unlike standard evolutionary algorithms, each Q-bit individual is a probability model, which can represent multiple solutions. Since probability models store global statistical information of good solutions found previously in the search, QEA have good potential to deal with hard optimization problems with many local optimal solutions. So far, not much work has been done on evolutionary multi-objective (EMO) algorithms with probabilistic representation. In this paper, we investigate the performance of two state-of-the-art EMO algorithms — MOEA/D and NSGA-II, with probabilistic representation based on pheromone trails, on the multi-objective travelling salesman problem. Our experimental results show that MOEA/D and NSGA-II with probabilistic presentation are very promising in sampling high-quality offspring solutions and in diversifying the search along the Pareto fronts.

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Evolutionary multi-objective optimization algorithms with probabilistic representation based on pheromone trails

This paper presents a novel algorithm to solve dynamic multiobjective optimization problems. In dynamic multiobjective optimization problems, multiple objective functions and/or constraints may change over time, which requires a multiobjective optimization algorithm to track the moving Pareto-optimal solutions and/or Pareto-optimal front. A first-order difference model is designed to predict the new locations of a certain number of Pareto-optimal solutions based on the previous locations when an environmental change is detected. In addition, a part of old Pareto-optimal solutions are retained to the new population. The prediction model is incorporated into a multiobjective evolutionary algorithm based on decomposition to solve the dynamic multiobjective optimization problems. In such a way, the changed POS or POF can be tracked more quickly. The proposed algorithm is tested on a number of typical benchmark problems with different dynamic characteristics and difficulties. Experimental results show that the proposed algorithm performs competitively when addressing dynamic multiobjective optimization problems in comparisons with the other state-of-the-art algorithms.

A First-Order Difference Model-Based Evolutionary Dynamic Multiobjective Optimization

This paper studies how to use two reproduction operators with different characteristics for balancing the convergence and the diversity in MOEA/D. We consider two operators. One is a differential evolution and polynomial mutation, and the other is a neighbor learning and inversion mutation. We show that these two operators have different search abilities. Then we propose a scheme to use these two operators in our recently proposed MOEA/D-GR framework. We test the proposed algorithm on some benchmark problems to demonstrate its effectiveness.

Balancing Convergence and Diversity by Using Two Different Reproduction Operators in MOEA/D: Some Preliminary Work

This paper introduces a differential prediction model to predict the varying Pareto-Optimal Solutions (POS) when solving dynamic multiobjective optimization problems (DMOPs). In dynamic multiobjective optimization problems, several competing objective functions and/or constraints change over time. As a consequence, the Pareto-Optimal Solutions and/or Pareto-Optimal Front may vary over time. The differential prediction model is used to forecast the shift vector in the decision space of the centroid in the population through the centroid's historical locations in three previous environments. This differential prediction model is incorporated into a multiobjective evolutionary algorithm based on decomposition to solve DMOPs. After detecting the environmental change, half of individuals in the population are forecasted their new positions in the decision space by using the differential prediction model and the others' positions are retained. The proposed model is tested on a number of typical benchmark problems with several dynamic characteristics. Experimental results show that the proposed model is competitively in comparisons with the other state-of-the-art models or approaches that were proposed for solving DMOPs.

Hui Li

Papers

A multi-phase multiobjective approach based on decomposition for sparse reconstruction

Evolutionary multi-objective optimization algorithms with probabilistic representation based on pheromone trails

A First-Order Difference Model-Based Evolutionary Dynamic Multiobjective Optimization

Balancing Convergence and Diversity by Using Two Different Reproduction Operators in MOEA/D: Some Preliminary Work

A differential prediction model for evolutionary dynamic multiobjective optimization