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The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link. Although evolutionary computing (EC) methods are stochastic optimization methods, it is usually difficult to find the global optimum by restarting the methods when the population converges to a local optimum. A major reason is that many optimization problems have basins of attraction (BoAs) that differ widely in shape and size, and the population always prefers to converge toward basins of attraction that are easy to search. Although heuristic restart based on tabu search is a theoretically feasible idea to solve this problem, existing EC methods with heuristic restart are difficult to avoid repetitive search results while maintaining search efficiency. This paper tries to overcome the dilemma by online learning the BoAs and proposes a search mode called history-guided hill exploration (HGHE). In the search mode, evaluated solutions are used to help separate the search space into hill regions which correspond to the BoAs, and a classical EC method is used to locate the optimum in each hill region. An instance algorithm for continuous optimization named HGHE differential evolution (HGHE-DE) is proposed to verify the effectiveness of HGHE. Experimental results prove that HGHE-DE can continuously discover unidentified BoAs and locate optima in identified BoAs.

The design of antenna arrays is usually imposed with lots of requirements. The design can be modeled as different kinds of optimization problems, such as single-objective optimization problems (SOPs) with or without constraints or multiobjective optimization problems with or without constraints. Then, different optimizers need to be applied to deal with the corresponding optimization problems. There are two issues to be addressed for antenna designers: one is which kind of optimization problems is the fittest to model antenna arrays; the other is a general and efficient algorithm that can solve different kinds of problems. To this end, for the first issue, constrained multiobjective optimization problems (CMOPs) are recommended as the most suitable kind in this article because CMOPs not only fully reflect the requirements of array design but also cover SOPs, constrained optimization problems (COPs), and multiobjective optimization problems (MOPs) as degeneration cases. For the second issue, a dynamic constrained multiobjective evolutionary algorithm (DCMOEA) is provided to solve CMOPs. Notably, DCMOEA is a general framework that can also work well on SOPs, COPs, and MOPs. Finally, two examples of antenna arrays are designed to verify the proposed conclusion.

The file attached to this record is the author's final peer reviewed version. In evolutionary algorithms, how to effectively select interactive solutions for generating offspring is a challenging problem. Though many operators are proposed, most of them select interactive solutions (parents) randomly, having no specificity for the features of landscapes in various problems. To address this issue, this paper proposes a reinforcement-learning-based evolutionary algorithm to select solutions within the approximated basin of attraction. In the algorithm, the solution space is partitioned by the k-dimensional tree, and features of subspaces are approximated with respect to two aspects: objective values and uncertainties. Accordingly, two reinforcement learning (RL) systems are constructed to determine where to search: the objective-based RL exploits basins of attraction (clustered subspaces) and the uncertainty-based RL explores subspaces that have been searched comparatively less. Experiments are conducted on widely used benchmark functions, demonstrating that the algorithm outperforms three other popular multimodal optimization algorithms.

The multi-population method is a common method for solving dynamic optimization problems. However, to design an efficient multi-population method, one of the challenging issues is how to allocate computational resources between populations given a limited computing buget in dynamic environments. This paper designs a contribution-based resource allocation mechanism. In this mechanism, a contribution degree of a population is defined according to the performance of the population, which determines the probability of the population to obtain the computing resource. This mechanism is implemented in an adaptive multi-population method. Experimental results on the moving peaks benchmark show that the algorithm equipped with the resource allocation mechanism outperforms the original algorithms.

This paper proposes a new method for handling the difficulty of multi-modality for the single-objective optimization problem (SOP). The method converts a SOP to an equivalent dynamic multi-objective optimization problem (DMOP). A new dynamic multi-objective evolutionary algorithm (DMOEA) is implemented to solve the DMOP. The DMOP has two objectives: the original objective and a niche-count objective. The second objective aims to maintain the population diversity for handling the multi-modality difficulty during the search process. Experimental results show that the performance of the proposed algorithm is significantly better than the state-of-the-art competitors on a set of benchmark problems and real world antenna array problems.

Balancing objectives optimization and constraints satisfaction are two equally important goals for constrained multi-objective optimization problems (CMOPs). However, most studies pay more attention on avoiding violating the constraints than achieving better objectives optimization. To alleviate this issue, this paper presents a dynamic constrained multi-objective evolutionary algorithm (DCMOEA) for handling constraints and optimizing objectives simultaneously. DCMOEA converts a m-objective COP to a dynamic (m+2)-objective COP. A simple yet effective dynamic niching technique is designed to enhance the population diversity from the decision space. An instantiation of the DCMOEA on NSGA-II (named DCNSGA-II) is implemented, and compared with five representative constraint-handling techniques on 26 well-known CMOPs. The experimental results indicate that the DCNSGA-II is highly competitive in solving CMOPs.

For computationally expensive constrained optimization problems, one crucial issue is that the existing expected improvement (EI) criteria are no longer applicable when a feasible point is not initially provided. To address this challenge, this paper uses the expected improvement of constraint violation to reach feasible region. A new constrained expected improvement criterion is proposed to select sample solutions for the update of Gaussian process (GP) surrogate models. The validity of the proposed constrained expected improvement criterion is proved theoretically. It is also verified by experimental studies and results show that it performs better than or competitive to compared criteria.

The typical formulation of a constrained optimization problem (COP), which has the constraint form of \(\mathbf {g}(\mathbf {x})\le \mathbf {0}\), has widely used in evolutionary computation field. However, it is not suitable for Gaussian processes (GPs) based expensive optimization. In this paper, a more general and suitable formulation of the COP, which has the constraint form of \(\mathbf {l_{g}} \le \mathbf {g}(\mathbf {x})\le \mathbf {u_{g}}\), is recommended for the expensive optimization. Modeling a real world COP as a typical formulation will probably introduce additional constraints and dependencies among the objective and constraints while that as a suitable one introduces none additional. In the case of typical formulation, the additional constraints and dependencies have to be handled and the handling would cost additional computational resource, especially, the additional dependencies would lead to degenerating the performance of expensive optimization technologies since most of the technologies are based on the assumption of mutual independency among the objective and constraints. Experiments show that the performance of the expensive optimization technologies in the aspect of precision on solving the problems with suitable formulation is better than that with typical one. However, we could not verify the expense of additional computational resource since there are few expensive optimization technologies dealing with dependent objective and constraints.

In differential evolution (DE) studies, there are many parameter adaptation methods, aiming at tuning the mutation factor $F$ and the crossover probability $\mathit {CR}$ . However, these methods still cannot resolve the issues of population premature convergence and population stagnation. To address these issues, in this paper, we investigate the population adaptation regarding population diversity at the dimensional level and propose a mechanism named auto-enhanced population diversity (AEPD) to automatically enhance population diversity. AEPD is able to identify the moments when a population becomes converging or stagnating by measuring the distribution of the population in each dimension. When convergence or stagnation is identified at a dimension, the population is diversified at that dimension to an appropriate level or to eliminate the stagnation issue. The AEPD mechanism was incorporated into a popular DE algorithm and it was tested on a set of 25 CEC2005 benchmark functions. The results showed that AEPD significantly improved the performance of the original algorithms. In addition, AEPD helped the algorithms become less sensitive to population size, a parameter widely considered problem dependent for many DE algorithms. The DE algorithm with AEPD also has a superior performance in comparison with several other peer algorithms.

A novel multiobjective technique is proposed for solving constrained optimization problems (COPs) in this paper. The method highlights three different perspectives: 1) a COP is converted into an equivalent dynamic constrained multiobjective optimization problem (DCMOP) with three objectives: a) the original objective; b) a constraint-violation objective; and c) a niche-count objective; 2) a method of gradually reducing the constraint boundary aims to handle the constraint difficulty; and 3) a method of gradually reducing the niche size aims to handle the multimodal difficulty. A general framework of the design of dynamic constrained multiobjective evolutionary algorithms is proposed for solving DCMOPs. Three popular types of multiobjective evolutionary algorithms, i.e., Pareto ranking-based, decomposition-based, and hype-volume indicator-based, are employed to instantiate the framework. The three instantiations are tested on two benchmark suites. Experimental results show that they perform better than or competitive to a set of state-of-the-art constraint optimizers, especially on problems with a large number of dimensions.

For the differential evolution (DE) algorithms, there are many parameter adaptation methods, which aim at tuning the mutation factor F and the crossover probability CR. When the population diversity is very small and has been converged in a local optimum, even if the evolution goes on, the population will no longer improve. This is also true for the DE algorithms with adaptive F and CR. The enhancement of population diversity is necessary to DE algorithms. In this paper, we test the JADE algorithm with auto-enhanced population diversity (AEPD) on the newest benchmark functions.

The multipopulation method has been widely used to solve dynamic optimization problems (DOPs) with the aim of maintaining multiple populations on different peaks to locate and track multiple changing optima simultaneously. However, to make this approach effective for solving DOPs, two challenging issues need to be addressed. They are how to adapt the number of populations to changes and how to adaptively maintain the population diversity in a situation where changes are complicated or hard to detect or predict. Tracking the changing global optimum in dynamic environments is difficult because we cannot know when and where changes occur and what the characteristics of changes would be. Therefore, it is necessary to take these challenging issues into account in designing such adaptive algorithms. To address the issues when multipopulation methods are applied for solving DOPs, this paper proposes an adaptive multi-swarm algorithm, where the populations are enabled to be adaptive in dynamic environments without change detection. An experimental study is conducted based on the moving peaks problem to investigate the behavior of the proposed method. The performance of the proposed algorithm is also compared with a set of algorithms that are based on multipopulation methods from different research areas in the literature of evolutionary computation.

In Differential Evolution (DE), there are many adaptive DE algorithms proposed for parameter adaptation. However, they mainly focus on tuning the mutation factor F and the crossover probability CR. The adaptation of population size NP has not been widely studied in the literature of DE. Reducing population size without jeopardizing the performance of an algorithm could save computational resources and hence accelerate it's convergence speed. This is beneficial to algorithms for optimization problems which need expensive evaluations. In this paper, we propose an improved population reduction method for DE, called dynNPMinD-DE, by considering the difference between individuals. When the reduction criterion is satisfied, dynNPMinD-DE selects the best individual and pairs of individuals with minimal-step difference vectors to form a new population. dynNPMinD-DE is tested on a set of 13 scalable benchmark functions in the number of dimensions of D=30 and D=50, respectively. The results show that dynNPMinD-DE outperforms the other peer DE algorithms in terms of both solution accuracy and convergence speed on most test functions.

In the last two decades, dynamic optimization problems (DOPs) have drawn a lot of research studies from the evolutionary computation (EC) community due to the importance in real-world applications. A variety of evolutionary computation approaches have been developed to address DOPs. In parallel with developing new approaches, many benchmark and real-world DOPs have been constructed and used to compare them under different performance measures. In this chapter, we describe the concept of DOPs and review existing dynamic test problems that are commonly used by researchers to investigate their EC approaches in the literature. Some discussions regarding the major features of existing dynamic test environments are presented. Typical dynamic benchmark problems and real-world DOPs are described in detail. We also review the performance measures that are widely used by researchers to evaluate and compare their developed EC approaches for DOPs. Suggestions are also given for potential improvement regarding dynamic test and evaluation environments for the EC community.

Maintaining population diversity is a crucial issue for the performance of evolutionary algorithms (EAs) in dynamic environments. In the literature of EAs for dynamic optimization problems (DOPs), many studies have been done to address this issue based on change detection techniques. However, many changes are hard or impractical to be detected in real-world applications. Although, some research has been done by means of maintaining diversity without change detection. These methods are not effective because the continuous focus on diversity slows down the optimization process. This paper presents a maintaining diversity method without change detection based on a clustering technique. The method was implemented through particle swarm optimization (PSO), which was named CPSOR. The performance of the CPSOR algorithm was evaluated on the GDBG benchmark. A comparison study with another algorithm based on change detection has shown the effectiveness of the CPSOR algorithm for tracking and locating the global optimum in dynamic environments.

This article is posted here with permission of the IEEE - Copyright @ 2009 IEEE Traditional particle swarm optimization (PSO) suffers from the premature convergence problem, which usually results in PSO being trapped in local optima. This paper presents an adaptive learning PSO (ALPSO) based on a variant PSO learning strategy. In ALPSO, the learning mechanism of each particle is separated into three parts: its own historical best position, the closest neighbor and the global best one. By using this individual level adaptive technique, a particle can well guide its behavior of exploration and exploitation. A set of 21 test functions were used including un-rotated, rotated and composition functions to test the performance of ALPSO. From the comparison results over several variant PSO algorithms, ALPSO shows an outstanding performance on most test functions, especially the fast convergence characteristic. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom under Grant EP/E060722/1.

This article is posted here with permission of the IEEE - Copyright @ 2009 IEEE In the real world, many applications are nonstationary optimization problems. This requires that optimization algorithms need to not only find the global optimal solution but also track the trajectory of the changing global best solution in a dynamic environment. To achieve this, this paper proposes a clustering particle swarm optimizer (CPSO) for dynamic optimization problems. The algorithm employs hierarchical clustering method to track multiple peaks based on a nearest neighbor search strategy. A fast local search method is also proposed to find the near optimal solutions in a local promising region in the search space. Six test problems generated from a generalized dynamic benchmark generator (GDBG) are used to test the performance of the proposed algorithm. The numerical experimental results show the efficiency of the proposed algorithm for locating and tracking multiple optima in dynamic environments. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom under Grant EP/E060722/1.

In the paper, for many-objective optimization problems, the authors pointed out that the Pareto Optimality is unfair, unreasonable and imperfect for Many-objective Optimization Problems (MOPs) underlying the hypothesis that all objectives have equal importance and propose a new evolutionary decision theory. The key contribution is the discovery of the new definition of optimality called E-optimality for MOP that is based on a new conception, so called E-dominance, which not only considers the difference of the number of superior and inferior objectives between two feasible solutions, but also considers the values of improved objective functions underlying the hypothesis that all objectives in the problem have equal importance. Two new evolutionary algorithms for E-optimal solutions are proposed. Because the new relation

This article is posted here with permission of IEEE - Copyright @ 2008 IEEE In the real world, many applications are non-stationary optimization problems. This requires that the optimization algorithms need to not only find the global optimal solution but also track the trajectory of the changing global best solution in a dynamic environment. To achieve this, this paper proposes a multi-swarm algorithm based on fast particle swarm optimization for dynamic optimization problems. The algorithm employs a mechanism to track multiple peaks by preventing overcrowding at a peak and a fast particle swarm optimization algorithm as a local search method to find the near optimal solutions in a local promising region in the search space. The moving peaks benchmark function is used to test the performance of the proposed algorithm. The numerical experimental results show the efficiency of the proposed algorithm for dynamic optimization problems.

This is a post-print version of the article - Copyright @ 2008 Springer-Verlag Evolutionary computation has become an important problem solving methodology among the set of search and optimization techniques. Recently, more and more different evolutionary techniques have been developed, especially hybrid evolutionary algorithms. This paper proposes an island based hybrid evolutionary algorithm (IHEA) for optimization, which is based on Particle swarm optimization (PSO), Fast Evolutionary Programming (FEP), and Estimation of Distribution Algorithm (EDA). Within IHEA, an island model is designed to cooperatively search for the global optima in search space. By combining the strengths of the three component algorithms, IHEA greatly improves the optimization performance of the three basic algorithms. Experimental results demonstrate that IHEA outperforms all the three component algorithms on the test problems. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant EP/E060722/1.

In this paper the authors point out that the Pareto Optimality is unfair, unreasonable and imperfect for Many-objective Optimization Problems (MOPs) underlying the hypothesis that all objectives have equal importance. The key contribution of this paper is the discovery of the new definition of optimality called Ɛ-optimality for MOP that is based on a new conception, so called Ɛ-dominance, which not only considers the difference of the number of superior and inferior objectives between two feasible solutions, but also considers the values of improved objective functions underlying the hypothesis that all objectives in the problem have equal importance. Two new evolutionary algorithms are given, where Ɛ- dominance is used as a selection strategy with the winning score as an elite strategy for search -optimal solutions. Two benchmark problems are designed for testing the new concepts of many-objective optimization problems. Numerical experiments show that the new definition of optimality is more perfect than that of the Pareto Optimality which is widely used in the evolutionary computation community for solving many-objective optimization problems.

Particle swarm optimization (PSO) has shown its fast search speed in many complicated optimization and search problems. However, PSO could often easily fall into local optima because the particles could quickly get closer to the best particle. At such situations, the best particle could hardly be improved. This paper proposes a new hybrid PSO (HPSO) to solve this problem by adding a Cauchy mutation on the best particle so that the mutated best particle could lead all the rest of particles to the better positions. Experimental results on many well-known benchmark optimization problems have shown that HPSO could successfully deal with those difficult multimodal functions while maintaining fast search speed on those simple unimodal functions in the function optimization

This paper presents a novel particle swarm optimiser (PSO) called PSO with simple and efficient neighbourhood search strategies (NSPSO), which employs one local and two global neighbourhood search strategies. By this way, one strong and two weak locality perturbation operators are embedded in the standard PSO. The NSPSO consists of two main steps. First, for each particle, three trail particles are generated by the mentioned three neighbourhood search strategies, respectively. Then, the best one among the three trail particles is selected to compete with the current particle, and the fitter one is accepted as a current particle. In order to verify the performance of NSPSO, it experimentally has been tested on 12 unimodal and multimodal benchmark functions. The results show that NPSO significantly outperforms other seven PSO variants.

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link. Real-time monitoring of drinking water in a water distribution system (WDS) can effectively warn and reduce safety risks. One of the challenges is to identify the contamination source through these observed data due to the real-time, non-uniqueness, and large scale characteristics. To address the real-time and non-uniqueness challenges, we propose an adaptive multi-population evolutionary optimization algorithm to determine the real-time characteristics of contamination sources, where each population aims to locate and track a different global optimum. The algorithm adaptively adjusts the number of populations using a feed-back learning mechanism. To effectively locate an optimal solution for a population, a co-evolutionary strategy is used to identify the location and the injection profile separately. Experimental results on three WDS networks show that the proposed algorithm is competitive in comparison with three other state-of-the-art evolutionary algorithms.