Yuren Zhou

A Vector Angle-Based Evolutionary Algorithm for Unconstrained Many-Objective Optimization

Taking both convergence and diversity into consideration, this paper suggests a vector angle-based evolutionary algorithm for unconstrained (with box constraints only) many-objective optimization problems. In the proposed algorithm, the maximum-vector-angle-first principle is used in the environmental selection to guarantee the wideness and uniformity of the solution set. With the help of the worse-elimination principle, worse solutions in terms of the convergence (measured by the sum of normalized objectives) are allowed to be conditionally replaced by other individuals. Therefore, the selection pressure toward the Pareto-optimal front is strengthened. The proposed method is compared with other four state-of-the-art many-objective evolutionary algorithms on a number of unconstrained test problems with up to 15 objectives. The experimental results have shown the competitiveness and effectiveness of our proposed algorithm in keeping a good balance between convergence and diversity. Furthermore, it was shown by the results on two problems from practice (with irregular Pareto fronts) that our method significantly outperforms its competitors in terms of both the convergence and diversity of the obtained solution sets. Notably, the new algorithm has the following good properties: 1) it is free from a set of supplied reference points or weight vectors; 2) it has less algorithmic parameters; and 3) the time complexity of the algorithm is low. Given both good performance and nice properties, the suggested algorithm could be an alternative tool when handling optimization problems with more than three objectives.

Enhancing the performance of differential evolution with covariance matrix self-adaptation ☆

Abstract Differential evolution (DE) is an efficient global optimizer, while the covariance matrix adaptation evolution strategy (CMA-ES) shows great power on local search. However, utilizing both of these advantages in one algorithm is difficult since the randomness introduced by DE may reduce the reliability of covariance matrix estimation. Moreover, the exploration ability of DE can be canceled out by CMA-ES because they use completely different mechanisms to control the search step. To take advantage of both DE and CMA-ES, we propose a novel DE variant with covariance matrix self-adaptation, named DECMSA. In DECMSA, a new mutation scheme named “DE/current-to-better/1” is implemented. This scheme uses a Gaussian distribution to guide the search and strengthens both exploration and exploitation capabilities of DE. The proposed algorithm has been tested on the CEC-13 benchmark suite. The experimental results demonstrate that DECMSA outperforms popular DE variants, and it is quite competitive with state-of-the-art CMA-ES variants such as IPOP-CMA-ES and BIPOP-CMA-ES. Moreover, equipped with a constraint handling method, DECMSA is able to produce better solutions than other comparative algorithms on three classic constrained engineering design problems.

An Evolution Path Based Reproduction Operator for Many-Objective Optimization

The many-objective evolutionary algorithms generally make use of a set of well-spread reference vectors to increase the selection pressure toward the Pareto front in high-dimensional objective space. However, few studies have been reported on how to generate new solutions toward the Pareto set (PS) in the decision space with the help of these reference vectors. To fill this gap, we develop a novel reproduction operator based on the differential evolution. The main idea is using the evolution paths to depict the population movement and predict its tendency. These evolution paths are used to create potential solutions, and thus, accelerate the convergence toward the PS. Furthermore, a self-adaptive mechanism is introduced to adapt related parameters automatically. This operator is implemented in two well-known many-objective evolutionary algorithm frameworks. The experimental results on 20 widely used benchmark problems show that the proposed operator is able to strengthen the performance of the original algorithms in handling many-objective optimization problems.

Configuring Software Product Lines by Combining Many-Objective Optimization and SAT Solvers

A feature model (FM) is a compact representation of the information of all possible products from software product lines. The optimal feature selection involves the simultaneous optimization of multiple (usually more than three) objectives in a large and highly constrained search space. By combining our previous work on many-objective evolutionary algorithm (i.e., VaEA) with two different satisfiability (SAT) solvers, this article proposes a new approach named SATVaEA for handling the optimal feature selection problem. In SATVaEA, an FM is simplified with the number of both features and constraints being reduced greatly. We enhance the search of VaEA by using two SAT solvers: one is a stochastic local search--based SAT solver that can quickly repair infeasible configurations, whereas the other is a conflict-driven clause-learning SAT solver that is introduced to generate diversified products. We evaluate SATVaEA on 21 FMs with up to 62,482 features, including two models with realistic values for feature attributes. The experimental results are promising, with SATVaEA returning 100% valid products on almost all FMs. For models with more than 10,000 features, the search in SATVaEA takes only a few minutes. Concerning both effectiveness and efficiency, SATVaEA significantly outperforms other state-of-the-art algorithms.

A many-objective evolutionary algorithm based on a projection-assisted intra-family election

Abstract In recent years, many researchers have put emphasis on the study of how to keep a good balance between convergence and diversity in many-objective optimization. This paper proposes a new many-objective evolutionary algorithm based on a projection-assisted intra-family election. In the proposed algorithm, basic evolution directions are adaptively generated according to the current population and potential evolution directions are excavated in each individuals family. Based on these evolution directions, a strategy of intra-family election is performed in every family and elite individuals are elected as representatives of the specific family to join the next stage, which can enhance the convergence of the algorithm. Moreover, a selection procedure based on angles is used to maintain the diversity. The performance of the proposed algorithm is verified and compared with several state-of-the-art many-objective evolutionary algorithms on a variety of well-known benchmark problems ranging from 5 to 20 objectives. Empirical results demonstrate that the proposed algorithm outperforms other peer algorithms in terms of both the diversity and the convergence of the final solutions set on most of the test instances. In particular, our proposed algorithm shows obvious superiority when handling the problems with larger number of objectives.

A historical solutions based evolution operator for decomposition-based many-objective optimization

Abstract As a kind of iterative algorithm based on population, an evolutionary algorithm generates many solutions at each generation. The historical solutions can provide information about the former generations and in turn help to guide the evolving of population. Thus, this paper utilizes the historical solutions and proposes a new evolution operator for decomposition-based many-objective optimization. The new operator combines the vertical information across different generations and the horizontal information from the current generation. Moreover, a two-stage bound-checking mechanism and an adaptive parameter setting scheme are designed to assist the proposed operator. After incorporating the proposed operator and some related techniques into the decomposition-based framework, we form a new algorithm called MOEA/D-HSE. The experimental results on well-known benchmark problems ranging from 5 to 15 objectives show that MOEA/D-HSE significantly outperforms other peer algorithms on the vast majority of the test instances, demonstrating the effectiveness and competitiveness of the proposed operator.

A local search based restart evolutionary algorithm for finding triple product property triples

As a mean to bound the exponent ω of the matrix multiplication, the group-theoretic approach to fast matrix multiplication was first introduced by Cohn and Umans in 2003. This involves a knotty problem, i.e., finding three subsets of a given group satisfying the so-called triple product property such that the product of their sizes is as large as possible. For this challenge problem, exact or randomized heuristic algorithms have been proposed, which are either time-consuming or ineffective on groups with large order. This paper proposes to use an evolutionary algorithm to solve the above problem. In the proposed algorithm, a local search and a restart strategy are employed to enhance the exploitation and exploration ability of the algorithm, respectively. The proposed approach is tested on a large number of nonabelian groups with order from 6 to 100. Experimental results show that the new algorithm can obtain a good tradeoff among effectiveness, robustness and efficiency. Especially, this approach is effective for nonabelian groups with order larger than 36, and obtains three subsets for each of these groups, which satisfy the triple product property and the product of whose sizes reaches the best found so for. Most importantly, we find by using the proposed algorithm 12 groups which would be promising to prove a nontrivial upper bound on the exponent ω of the matrix multiplication.