Bo-Yang Qu

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

Abstract 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.

Real-parameter evolutionary multimodal optimization — A survey of the state-of-the-art

Abstract Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space and as and when needed, the current solution may be switched to another suitable one while still maintaining the optimal system performance. Evolutionary Algorithms (EAs), due to their population-based approaches, are able to detect multiple solutions within a population in a single simulation run and have a clear advantage over the classical optimization techniques, which need multiple restarts and multiple runs in the hope that a different solution may be discovered every run, with no guarantee however. Numerous evolutionary optimization techniques have been developed since late 1970s for locating multiple optima (global or local). These techniques are commonly referred to as “niching” methods. Niching can be incorporated into a standard EA to promote and maintain formation of multiple stable subpopulations within a single population, with an aim to locate multiple globally optimal or suboptimal solutions simultaneously. This article is the first of its kind to present a comprehensive review of the basic concepts related to real-parameter evolutionary multimodal optimization, a survey of the major niching techniques, a detailed account of the adaptation of EAs from diverse paradigms to tackle multimodal problems, benchmark problems and performance measures.

A Distance-Based Locally Informed Particle Swarm Model for Multimodal Optimization

Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space and when needed, the current solution may be switched to a more suitable one while still maintaining the optimal system performance. Niching particle swarm optimizers (PSOs) have been widely used by the evolutionary computation community for solving real-parameter multimodal optimization problems. However, most of the existing PSO-based niching algorithms are difficult to use in practice because of their poor local search ability and requirement of prior knowledge to specify certain niching parameters. This paper has addressed these issues by proposing a distance-based locally informed particle swarm (LIPS) optimizer, which eliminates the need to specify any niching parameter and enhance the fine search ability of PSO. Instead of using the global best particle, LIPS uses several local bests to guide the search of each particle. LIPS can operate as a stable niching algorithm by using the information provided by its neighborhoods. The neighborhoods are estimated in terms of Euclidean distance. The algorithm is compared with a number of state-of-the-art evolutionary multimodal optimizers on 30 commonly used multimodal benchmark functions. The experimental results suggest that the proposed technique is able to provide statistically superior and more consistent performance over the existing niching algorithms on the test functions, without incurring any severe computational burdens.

Differential Evolution With Neighborhood Mutation for Multimodal Optimization

In this paper, a neighborhood mutation strategy is proposed and integrated with various niching differential evolution (DE) algorithms to solve multimodal optimization problems. Although variants of DE are highly effective in locating a single global optimum, no DE variant performs competitively when solving multi-optima problems. In the proposed neighborhood based differential evolution, the mutation is performed within each Euclidean neighborhood. The neighborhood mutation is able to maintain the multiple optima found during the evolution and evolve toward the respective global/local optimum. To test the performance of the proposed neighborhood mutation DE, a total of 29 problem instances are used. The proposed algorithms are compared with a number of state-of-the-art multimodal optimization approaches and the experimental results suggest that although the idea of neighborhood mutation is simple, it is able to provide better and more consistent performance over the state-of-the-art multimodal algorithms. In addition, a comparative survey on niching algorithms and their applications are also presented.

Multi-objective evolutionary algorithms based on the summation of normalized objectives and diversified selection

Most multi-objective evolutionary algorithms (MOEAs) use the concept of dominance in the search process to select the top solutions as parents in an elitist manner. However, as MOEAs are probabilistic search methods, some useful information may be wasted, if the dominated solutions are completely disregarded. In addition, the diversity may be lost during the early stages of the search process leading to a locally optimal or partial Pareto-front. Beside this, the non-domination sorting process is complex and time consuming. To overcome these problems, this paper proposes multi-objective evolutionary algorithms based on Summation of normalized objective values and diversified selection (SNOV-DS). The performance of this algorithm is tested on a set of benchmark problems using both multi-objective evolutionary programming (MOEP) and multi-objective differential evolution (MODE). With the proposed method, the performance metric has improved significantly and the speed of the parent selection process has also increased when compared with the non-domination sorting. In addition, the proposed algorithm also outperforms ten other algorithms.

Niching particle swarm optimization with local search for multi-modal optimization

Multimodal optimization is still one of the most challenging tasks for evolutionary computation. In recent years, many evolutionary multi-modal optimization algorithms have been developed. All these algorithms must tackle two issues in order to successfully solve a multi-modal problem: how to identify multiple global/local optima and how to maintain the identified optima till the end of the search. For most of the multi-modal optimization algorithms, the fine-local search capabilities are not effective. If the required accuracy is high, these algorithms fail to find the desired optima even after converging near them. To overcome this problem, this paper integrates a novel local search technique with some existing PSO based multimodal optimization algorithms to enhance their local search ability. The algorithms are tested on 14 commonly used multi-modal optimization problems and the experimental results suggest that the proposed technique not only increases the probability of finding both global and local optima but also reduces the average number of function evaluations.

Novel multimodal problems and differential evolution with ensemble of restricted tournament selection

Multi-modal optimization refers to locating not only one optimum but a set of locally optimal solutions. Niching is an important technique to solve multi-modal optimization problems. The ability of discover and maintain multiple niches is the key capability of these algorithms. In this paper, differential evolution with an ensemble of restricted tournament selection (ERTS-DE) algorithm is introduced to perform multimodal optimization. The algorithms is tested on 15 newly designed scalable benchmark multi-modal optimization problems and compared with the crowding differential evolution (Crowding-DE) in the literature. As shown by the experimental results, the proposed algorithm outperforms the Crowding-DE on the novel scalable benchmark problems.

Multi-objective evolutionary programming without non-domination sorting is up to twenty times faster

In this paper, Multi-objective evolutionary programming (MOEP) using fuzzy rank-sum with diversified selection is introduced. The performances of this algorithm as well as MOEP with non-domination sorting on the set of benchmark functions provided for CEC2009 Special Session and competition on Multi-objective Optimization are reported. With this rank-sum sorting and diversified selection, the speed of the algorithm has increased significantly, in particular by about twenty times on five objective problems when compared with the implementation using the non-domination sorting. Beside this, the proposed approach has performed either comparable or better than the MOEP with non-domination sorting.

Novel benchmark functions for continuous multimodal optimization with comparative results

Abstract Multi-modal optimization is concerned with locating multiple optima in one single run. Finding multiple solutions to a multi-modal optimization problem is especially useful in engineering, as the best solution may not always be the best realizable due to various practical constraints. To compare the performances of multi-modal optimization algorithms, multi-modal benchmark problems are always required. In this paper, 15 novel scalable multi-modal and real parameter benchmark problems are proposed. Among these 15 problems, 8 are extended simple functions while the rest are composition functions. These functions coordinate rotation and shift operations to create linkage among different dimensions and to place the optima at different locations, respectively. Four typical niching algorithms are used to solve the proposed problems. As shown by the experimental results, the proposed problems are challenging to these four recent algorithms.

Economic emission dispatch problems with stochastic wind power using summation based multi-objective evolutionary algorithm

In recent years, renewable energy sources such as wind energy have been used as one of the most effective ways to reduce pollution emissions. In this paper, a summation based multi-objective differential evolution (SMODE) algorithm is used to optimize the economic emission dispatch problem with stochastic wind power. The Weibull probability distribution function is used to model the stochastic nature of the wind power and the uncertainty is treated as the system constraints with stochastic variables. The algorithm is integrated with the superiority of feasible solution constraint handling technique. To validate the effectiveness of the proposed method, the standard IEEE 30-bus 6-generator test system with wind power (with/without considering losses) is studied with fuel cost and emission as two conflicting objectives to be optimized at the same time. Besides, a larger 40-generator system with wind farms is also solved by the proposed method. The results generated by SMODE are compared with those obtained using NSGAII as well as a number of techniques reported in literature. The results reveal that SMODE generates superior and consistent solutions.