Adaptive differential evolution algorithm for multiobjective optimization problems

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Abstract

In this paper, a new adaptive differential evolution algorithm (ADEA) is proposed for multiobjective optimization problems. In ADEA, the variable parameter F based on the number of the current Pareto-front and the diversity of the current solutions is given for adjusting search size in every generation to find Pareto solutions in mutation operator, and the select operator combines the advantages of DE with the mechanisms of Pareto-based ranking and crowding distance sorting. ADEA is implemented on five classical multiobjective problems, the results illustrate that ADEA efficiently achieves two goals of multiobjective optimization problems: find the solutions converge to the true Pareto-front and uniform spread along the front.

Introduction

Multiobjective optimization problems (MOPs) which consist of several competing and incommensurable objective functions are frequently encountered in real-world problem such as scientific and engineering applications. Consequently, there are two goals in multiobjective optimization: (i) to discover solutions as close to the Pareto-front as possible, (ii) to find solutions as diverse as possible in the obtained nondominated front. In recent years, many optimization techniques have been proposed in some literatures to solve MOPs. Some of the most attractive algorithms are evolution algorithms (EAs) such as NSGAII [1], MODE [2], PAES [3]. In contrast to traditional gradient-based techniques, EAs use a set of potential solutions to detect feasible region. So several solutions of a multiobjective problem can be obtained in a single run. The properties enable EAs converge fast to the true Pareto-front (the concept will be explained in the following section). In MOPs, a large number of optimal solutions exist, and each correspond to a different trade-off among the objective functions.

Differential evolution algorithm (DE) is designed for minimizing functions of real variable. It is extremely robust in locating the global minimum. DE is a simple yet powerful evolutionary optimization algorithm that has been successfully used in solving single-objective problems by Price and Storn [4]. After that, it was used to handle MOPs. Abbass was the first to apply DE to MOPs in the so-called Pareto differential evolution (PDE) algorithm [5], [6]. Madavan achieved good results with the Pareto differential evolution approach (PDEA) [7]. Xue introduced multiobjective differential evolution [2]. Tea Robicˆ proposed differential evolution for multiobjective optimization (DEMO) [8].

In this paper, we propose a new way of extending DE to be suitable for solving MOPs. A novel approach, called adaptive differential evolution algorithm (ADEA) which incorporated a new select operator and an adaptive parameter, is introduced to search the global solutions. In each generation, the select operator emphasize Elitist to promote the research towards the Pareto-front. And the adaptive parameter F was used to adjust step size for the need of algorithm. From the simulate results on five test problems, we find the speed of converging to the true Pareto-front of ADEA is more fast and the diversity is better than most of other optimization algorithms for multiobjective problems.

The rest of the paper is organized as follows: Section 2 provides the concept of Pareto-front and Pareto-ranking, DE scheme which was used as a background for ADEA. In Section 3, a method called ADEA is discussed in detail. Section 4 outlines the applied test problems and performance measures. Experimental results showing the effectiveness of our approach and the further comparison and discussion of the results are also provided in this section. Section 5 concludes the paper.

Section snippets

Pareto-ranking

Two or more objectives in a multiobjective problem are usually conflict and compete, they cannot be minimized concurrently. So one solution often cannot be said to be better or worse than another according their function values. There exists not a single optimal solution, but a set optimal solutions-called Pareto-front.

Starting from a set of initial solutions, multiobjective evolutionary algorithms use an iteratively improving optimization techniques to find the optimal solutions. In every

Adaptive differential evolution algorithm

When applying DE to MOPs, we face many difficulties. How to replace the parent with trial solutions and preserve a uniform spread of nondominated solutions are two main challenge for DE algorithm. The selection operator of original DE is simple but it is not fit for the MOPs. The selection is easy in single-objective optimization, but the selection is not so straight forward in MOPs. It exists that the trial solutions and the parent solutions are incomparable. According to the original DE, if

Performance measures

To validate our approach, we used the methodology normally adopted in the evolutionary multiobjective optimization literature. Because we wanted to compare ADEA to other MOEAs on their published results, we use three metrics that have been used in these studies. They represent both quantitative comparisons and qualitative comparisons with MOEAs that are respective of the state-of-the-art: the Nondominated Sorting Genetic Algorithm II (NSGAII) the Strength Pareto Evolutionary Algorithm (SPEA),

Conclusion

ADEA is a new DE implementation dealing with multiple objectives. The biggest difference between ADEA and other MOEAs is that ADEA introduced a new defined self-parameter and a new select operator. We tested the approach on five benchmark problems and it was found that our approach is competitive to most other approaches. We also experimented with different crossover rate on these problems to find their best solutions. The crossover rate is found to be sensitive on problem4 to the solutions. In

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