An enhanced-indicator based many-objective evolutionary algorithm with adaptive reference point

Abstract

Indicator based many-objective evolutionary algorithms generally introduce the performance indicator as the selection criterion in environmental selection. In the calculation of some indicators, the reference points as sampled points on Pareto fronts are very important for their calculation. However, it is difficult to obtain good reference points on various types of Pareto fronts. To address this issue, this paper proposes an enhanced-indicator based many-objective evolutionary algorithm with adaptive reference point, termed EIEA. The algorithm proposes a reference point adaptation method to dynamically adapt the reference points for the calculation of indicators. Moreover, the calculation of IGD-NS is enhanced by employing the modified distance calculation to introduce the Pareto compliant which can further comprehensively measure the convergence and diversity. The proposed EIEA adopts Pareto dominance and the enhanced IGD-NS as the first selection criterion and the secondary selection criterion in environmental selection, respectively. The intensive experiments demonstrate that the proposed algorithm has good performance in solving problems with various types of Pareto fronts, surpassing several representative many-objective evolutionary algorithms for many-objective optimization.

Introduction

Many real-world problems involve more than one conflicting objectives to be optimized simultaneously. These problems are referred to as multi-objective optimization problems (MOPs) [1,2], which can be stated as follows:

$$\begin{array}{l}\text{minimize }F\left(\mathbf{x}\right)={\left(f_1\left(\mathbf{x}\right),\dots ,f_M\left(\mathbf{x}\right)\right)}^T\\ \text{subject to }\mathbf{x}\in \Omega ,\end{array}$$

where $\Omega \subseteq {\mathbf{R}}^n$ is the decision space with $\mathbf{x}={(x_1,\dots ,x_n)}^T\in \Omega$ being the decision vector, $F:\Omega \to Y\subseteq {\mathbf{R}}^M$ consists of M objectives and Y is the objective space.

In the past three decades, a lot of multi-objective evolutionary algorithms (MOEAs) have been proposed to solve MOPs. These MOEAs have shown promising performance in solving MOPs [3]. However, some recent studies have demonstrated that conventional MOEAs faced with difficulties when handling MOPs with more than three objectives, and these MOPs are further differentiated as many-objective optimization problems (MaOPs) [4,5]. Due to the increasing number of objectives, traditional MOEAs lack selection pressure when solving MaOPs. To enhance the performance of MOEAs in solving MaOPs, a variety of many-objective evolutionary algorithms (MaOEAs) have been proposed. In dominance relation based methods, the conventional Pareto-based methods face difficulties to distinguish solutionsin tackling MaOPs. The most straightforward method is to relax the domination relation to enhance the selection pressure towards the Pareto front (PF) [[6], [7], [8]]. There are also other novel strategies to enhance convergence. Recently, Zhang et al. proposes a knee point driven evolutionary algorithm (KnEA) [9], which introduces a knee point based selection criterion to enhance convergence.

Unlike dominance based MaOEAs, decomposition based MaOEAs provide a promising approach to solve MaOPs [10]. The multiobjective evolutionary algorithm based on decomposition (MOEA/D) [11] is a representative of this class of algorithms. Recently, R. Cheng et al. propose a reference vector guided evolutionary algorithm (RVEA) for MaOPs [12], which decomposes MaOPs into a set of single-objective optimization problems (SOPs) via a set of uniformly distributed reference vectors, such that the candidate solutions can efficiently converge to the optimum of each SOP without considering the conflicts between different objectives. An evolutionary many-objective optimization algorithm using reference point based nondominated sorting approach (NSGA-III) is proposed in Refs. [13], which adopts the reference points based secondary criterion to manage diversity. In Ref. [14], a dynamical decomposition based evolutionary algorithm (DDEA) is proposed to solve MaOPs, which introduces a dynamical decomposition strategy to partition the objective space by solutions themselves without predefined reference vectors.

Indicator-based MaOEAs provide another potential approach to solve MaOPs. Their core idea is to adopt performance indicators of solutions quality measurement as selection criteria in environmental selection [15]. R2 indicator based many-objective metaheuristic-II (MOMBI-II) is proposed in Ref. [16], which proposes a reference point update method to solve the problem that the diversity of the approximation sets is sensitive to the choice of the reference points during normalization. In Ref. [17], an IGD-NS (enhanced inverted generational distance) indicator based evolutionary algorithm, termed AR-MOEA, which proposes a reference point adaptation to handle different types of Pareto fronts. An IGD indicator-based evolutionary algorithm (MaOEA/IGD) is proposed by Yanan Sun et al. [18], which introduces the IGD indicator as selection criteria to select solutions in each generation, and one rank assignment mechanism is proposed to compare the dominance relation of the solutions with the reference points.

These MaOEAs are proposed to solve MaOPs, and the most existing MaOEAs have shown potential performance in solving different types of MaOPs. However, some recent studies point out that the most existing MaOEAs always encounter difficulties in diversity management for many-objective optimization [19,20]. Hence, this paper proposes an enhanced-indicator based many-objective evolutionary algorithm with adaptive reference point, namely EIEA. The major contributions of this paper are summarized as follows:

• The modified distance calculation in Ref. [21] is introduced to enhance the calculation of IGD-NS indicator. Compared with IGD-NS, the enhanced IGD-NS (IGD-NS⁺) can not only comprehensively measure population but also have Pareto compliant.

• A reference point adaptation strategy is proposed for many-objective optimization. This method can adjust the distribution of reference points to adapt different types of Pareto fronts.

• The effectiveness of the proposed reference point adaptation method is assessed via comparisons with two reference point adaptation methods. It demonstrates that the proposed method has good performance for adapting different shapes of Pareto fronts.

The remainder of this paper is organized as follows. In section 2, related work of this paper is presented. The details of the proposed algorithm EIEA are described in section 3. The empirical results of EIEA compared with several representative MaOEAs are presented in section 4. Finally, section 5 concludes this paper.