Zefeng Chen

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.

A Decomposition-Based Many-Objective Artificial Bee Colony Algorithm

In this paper, a decomposition-based artificial bee colony (ABC) algorithm is proposed to handle many-objective optimization problems (MaOPs). In the proposed algorithm, an MaOP is converted into a number of subproblems which are simultaneously optimized by a modified ABC algorithm. The hybrid of the decomposition-based algorithm and the ABC algorithm can make full use of the advantages of both algorithms. The former, with the help of a set of weight vectors, is able to maintain a good diversity among solutions, while the latter, with a fast convergence speed, is highly effective when solving a scalar optimization problem. Therefore, the convergence and diversity would be well balanced in the new algorithm. Moreover, subproblems in the proposed algorithm are handled unequally, and computational resources are dynamically allocated through specially designed onlooker bees and scout bees. The proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on 13 test problems with up to 50 objectives. It is shown by the experimental results that the proposed algorithm performs better than or comparably to other algorithms in terms of both quality of the final solution set and efficiency of the algorithms. Finally, as shown by the Wilcoxon signed-rank test results, the onlooker bees and scout bees indeed contribute to performance improvements of the algorithm. Given the high quality of solutions and the rapid running speed, the proposed algorithm could be a promising tool when approximating a set of well-converged and properly distributed nondominated solutions for MaOPs.

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.

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 Scalar Projection and Angle-Based Evolutionary Algorithm for Many-Objective Optimization Problems

In decomposition-based multiobjective evolutionary algorithms, the setting of search directions (or weight vectors), and the choice of reference points (i.e., the ideal point or the nadir point) in scalarizing functions, are of great importance to the performance of the algorithms. This paper proposes a new decomposition-based many-objective optimizer by simultaneously using adaptive search directions and two reference points. For each parent, binary search directions are constructed by using its objective vector and the two reference points. Each individual is simultaneously evaluated on two fitness functions—which are motivated by scalar projections—that are deduced to be the differences between two penalty-based boundary intersection (PBI) functions, and two inverted PBI functions, respectively. Solutions with the best value on each fitness function are emphasized. Moreover, an angle-based elimination procedure is adopted to select diversified solutions for the next generation. The use of adaptive search directions aims at effectively handling problems with irregular Pareto-optimal fronts, and the philosophy of using the ideal and nadir points simultaneously is to take advantages of the complementary effects of the two points when handling problems with either concave or convex fronts. The performance of the proposed algorithm is compared with seven state-of-the-art multi-/many-objective evolutionary algorithms on 32 test problems with up to 15 objectives. It is shown by the experimental results that the proposed algorithm is flexible when handling problems with different types of Pareto-optimal fronts, obtaining promising results regarding both the quality of the returned solution set and the efficiency of the new algorithm.

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.