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

A multiobjective optimization problem involves several conflicting objectives and has a set of Pareto optimal solutions. By evolving a population of solutions, multiobjective evolutionary algorithms (MOEAs) are able to approximate the Pareto optimal set in a single run. MOEAs have attracted a lot of research effort during the last 20 years, and they are still one of the hottest research areas in the field of evolutionary computation. This paper surveys the development of MOEAs primarily during the last eight years. It covers algorithmic frameworks such as decomposition-based MOEAs (MOEA/Ds), memetic MOEAs, coevolutionary MOEAs, selection and offspring reproduction operators, MOEAs with specific search methods, MOEAs for multimodal problems, constraint handling and MOEAs, computationally expensive multiobjective optimization problems (MOPs), dynamic MOPs, noisy MOPs, combinatorial and discrete MOPs, benchmark problems, performance indicators, and applications. In addition, some future research issues are also presented.

Multiobjective evolutionary algorithms: A survey of the state of the art

Grasshopper Optimisation Algorithm

Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition Qingfu Zhang∗, Aimin Zhou∗, Shizheng Zhao†, Ponnuthurai Nagaratnam Suganthan†, Wudong Liu∗and Santosh Tiwari‡ ∗Technical Report CES-487 The School of Computer Science and Electronic Engieering University of Essex, Colchester, C04, 3SQ, UK †School of Electrical and Electronic Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore ‡Department of Mechanical Engineering Clemson University, Clemson, SC 29634, US

Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition

In this paper, the performance of dynamic multi-swarm particle swarm optimizer (DMS-PSO) on the set of benchmark functions provided for the CEC2008 Special Session on Large Scale optimization is reported Different from the existing multi-swarm PSOs and local versions of PSO, the sub-swarms are dynamic and the sub-swarmspsila size is very small The whole population is divided into a large number sub-swarms, these sub-swarms are regrouped frequently by using various regrouping schedules and information is exchanged among the particles in the whole swarm The Quasi-Newton method is combined to improve its local searching ability

Dynamic multi-swarm particle swarm optimizer with local search for Large Scale Global Optimization

The multiobjective evolutionary algorithm based on decomposition (MOEA/D) has demonstrated superior performance by winning the multiobjective optimization algorithm competition at the CEC 2009. For effective performance of MOEA/D, neighborhood size (NS) parameter has to be tuned. In this letter, an ensemble of different NSs with online self-adaptation is proposed (ENS-MOEA/D) to overcome this shortcoming. Our experimental results on the CEC 2009 competition test instances show that an ensemble of different NSs with online self-adaptation yields superior performance over implementations with only one fixed NS.

Decomposition-Based Multiobjective Evolutionary Algorithm With an Ensemble of Neighborhood Sizes

In this paper, self-adaptive differential evolution (DE) is enhanced by incorporating the JADE mutation strategy and hybridized with modified multi-trajectory search (MMTS) algorithm (SaDE-MMTS) to solve large-scale continuous optimization problems. The JADE mutation strategy, the “DE/current-to-pbest” which is a variation of the classic “DE/current-to-best”, is used for generating mutant vectors. After the mutation phase, the binomial (uniform) crossover, the exponential crossover as well as no crossover option are used to generate each pair of target and trial vectors. By utilizing the self-adaptation in SaDE, both trial vector generation strategies and their associated control parameter values are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, suitable offspring generation strategy along with associated parameter settings will be determined adaptively to match different phases of the search process. MMTS is applied frequently to refine several diversely distributed solutions at different search stages satisfying both the global and the local search requirement. The initialization of step sizes is also defined by a self-adaption during every MMTS step. The success rates of both SaDE and the MMTS are determined and compared; consequently, future function evaluations for both search algorithms are assigned proportionally to their recent past performance. The proposed SaDE-MMTS is employed to solve the 19 numerical optimization problems in special issue of soft computing on scalability of evolutionary algorithms for large-scale continuous optimization problems and competitive results are presented.

Shi-Zheng Zhao

Papers

Multiobjective evolutionary algorithms: A survey of the state of the art

Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition

Dynamic multi-swarm particle swarm optimizer with local search for Large Scale Global Optimization

Decomposition-Based Multiobjective Evolutionary Algorithm With an Ensemble of Neighborhood Sizes

Self-adaptive differential evolution with multi-trajectory search for large-scale optimization