Shi-Zheng Zhao

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.

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

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.

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

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.

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

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.

Two-lbests based multi-objective particle swarm optimizer

The global best (gbest) or local best (lbest) of every particle in state-of-the-art multi-objective particle swarm optimization (MOPSO) implementations is selected from the non-dominated solutions in the external archive. This approach emphasizes the elitism at the expense of diversity when the size of the current set of non-dominated solutions in the external archive is small. This article proposes that the gbests or lbests should be chosen from the top fronts in a non-domination sorted external archive of reasonably large size. In addition, a novel two local bests (lbest) based MOPSO (2LB-MOPSO) version is proposed to focus the search around small regions in the parameter space in the vicinity of the best existing fronts unlike the current MOPSO variants in which the pbest and gbest (or lbest) of a particle can be located far apart in the parameter space thereby potentially resulting in a chaotic search behaviour. Comparative evaluation using 19 multi-objectives test problems and 11 state-of-the-art multi-objective evolutionary algorithms ranks overall the 2LB-MOPSO as the best while two state-of-the-art MOPSO algorithms are ranked the worst with respect to other multi-objective evolutionary algorithms.

Dynamic multi-swarm particle swarm optimizer with harmony search

In this paper, the dynamic multi-swarm particle swarm optimizer (DMS-PSO) is improved by hybridizing it with the harmony search (HS) algorithm and the resulting algorithm is abbreviated as DMS-PSO-HS. We present a novel approach to merge the HS algorithm into each sub-swarm of the DMS-PSO. Combining the exploration capabilities of the DMS-PSO and the stochastic exploitation of the HS, the DMS-PSO-HS is developed. The whole DMS-PSO population is divided into a large number of small and dynamic sub-swarms which are also individual HS populations. These sub-swarms are regrouped frequently and information is exchanged among the particles in the whole swarm. The DMS-PSO-HS demonstrates improved on multimodal and composition test problems when compared with the DMS-PSO and the HS.

Multi-objective robust PID controller tuning using two lbests multi-objective particle swarm optimization

In this paper, two lbests multi-objective particle swarm optimization (2LB-MOPSO) is applied to design multi-objective robust Proportional-integral-derivative (PID) controllers for two MIMO systems, namely, distillation column plant and longitudinal control system of the super maneuverable F18/HARV fighter aircraft. Multi-objective robust PID controller design problem is formulated by minimizing integral squared error (ISE) and balanced robust performance criteria. During the search, 2LB-MOPSO can focus on small regions in the parameter space in the vicinity of the best existing fronts. As the lbests are chosen from the top fronts in a non-domination sorted external archive of reasonably large size, the offspring obtained can be more diverse with good fitness. The performance of various optimal PID controllers is compared in terms of the sum of ISE and balanced robust performance criteria. For the purpose of comparison, 2LB-MOPSO, NSGA-II as well as earlier reported Riccati, IGA and OSA methods are considered. The performance of PID controllers obtained using 2LB-MOPSO is better than that of others. In addition, Hypervolume-based comparisons are carried out to show the superior performance of 2LB-MOPSO over NSGA-II. The results reveal that 2LB-MOPSO yields better robustness and consistency in terms of the sum of ISE and balanced robust performance criteria than various optimal PID controllers.

Empirical investigations into the exponential crossover of differential evolutions

Abstract Since the introduction in 1995, Differential evolution (DE) has drawn the attention of many researchers all over the world. Even though binomial (uniform) and exponential (modular two-point) crossover operators have been proposed simultaneously, binomial crossover has been more frequently used. Recently, the exponential crossover operator demonstrated superior performance over the binomial operator when solving complex high dimensional problems. We observe that the commonly used definition of exponential crossover operator does not scale impractically with the dimensionality of the problem being solved. Motivated by these observations, in this research, we investigate the performance of the current exponential crossover operator (EXP) and demonstrate its deficiencies. Consequently, a linearly scalable exponential crossover operator (LS-EXP) based on a number of consecutive dimensions to crossover is defined. Our numerical results on the most recent benchmark problems with dimensions ranging from 50 to 1000 show superior performance of the LS-EXP over the current exponential crossover operator, EXP.

Artificial foraging weeds for global numerical optimization over continuous spaces

Invasive Weed Optimization (IWO) is a recently developed derivative-free metaheuristic algorithm that mimics the robust process of weeds colonization and distribution in an ecosystem. On the other hand central to an ecosystem is the foraging behavior that pertains to the act of searching for food and forms an integral part of the daily life of most of the living creatures. For over past two decades, a few significant optimization algorithms were developed by emulating the foraging behavior of creatures like ants, bacteria, fish, bees etc. This article presents a hybrid real-parameter optimizer developed by incorporating the principles of Optimal Foraging Theory (OFT) in IWO, with a view to improving the search mechanism of the latter over discontinuous and multi-modal fitness landscapes, riddled with local optima. The hybridization does not impose any serious computational burden on IWO in terms of increasing number of Function Evaluations (FEs). The performance of the resulting hybrid algorithm has been compared with eleven other state-of-the-art metaheuristic algorithms over a test-suite of 16 numerical benchmarks taken from the CEC (Congress on Evolutionary Computation) 2005 competition and special session on real parameter optimization. Our simulation experiments indicate that the proposed algorithm is able to attain comparable results against the nine other optimizers. Owing to its promising performance on benchmarks and ease of implementation (without requiring much programming overhead), the proposed algorithm may serve as an attractive alternative for a plethora of practical optimization problems.

Comprehensive comparison of convergence performance of optimization algorithms based on nonparametric statistical tests

In evolutionary computation, statistical tests are commonly used to improve the comparative evaluation process of the performance of different algorithms. In this paper, three state-of-the-art Differential Evolution (DE) based algorithms, namely Dynamic Memetic Differential Evolution (MOS), Self-adaptive DE hybridized with modified multi-trajectory search (MMTS) algorithm (SaDE-MMTS) and Self-adaptive Differential Evolution Algorithm using Population Size Reduction and three Strategies Algorithm (jDElscop) as well as a novel algorithm called ensemble of parameters and mutation strategies in Differential Evolution with Self-adaption and MMTS (Sa-EPSDE-MMTS), are tested on the most recent LSO benchmark problems and comparatively evaluated using nonparametric statistical analysis. Instead of using the “Value-to-Reach” as the comparison criterion, comprehensive comparison over multiple evolution points are investigated on each test problem in order to quantitatively compare convergence performance of different algorithms. Our investigations demonstrate that even though all these algorithms yield the same final solutions on a large set of problems, they possess statistically significant variations during the convergence. Hence, we propose that evolutionary algorithms can be compared statistically along the evolution paths.