Elsevier

Information Sciences

Volume 626, May 2023, Pages 658-693
Information Sciences

A strength pareto evolutionary algorithm based on adaptive reference points for solving irregular fronts

https://doi.org/10.1016/j.ins.2023.01.073Get rights and content

Abstract

Evolutionary algorithms have proven to be extremely effective at tackling multi-objective optimization problems (MOPs). However, when dealing with many-objective optimization problems (MaOPs), their performance frequently degrades, especially when the Pareto marking irregular shapes. The population pressure to choose the Pareto optimal front and address the generalizability of different Pareto front shapes becomes more challenging as the number of objectives increases. We present a strength Pareto evolutionary algorithm based on adaptive reference points (SPEA/ARP) to address this problem. First, the reference points are updated using current and historical population information. The angles between the current demographic information and the predefined uniform reference points are used to select the active reference points, and the adaptive reference points are selected from the historical population information projected onto the reference plane. Second, the fitness function values are applied to classify the environmental selection criteria into two categories: 1) The angle distance scaling function using adaptive reference points is utilized to increase selection pressure, and the diversity of non-dominated solutions is balanced using the angle-based secondary selection technique. 2) Otherwise, the fitness function values are employed to choose the next generation of non-dominated solutions. Third, an aggregate fitness r-value generated by the angle distance scaling function is employed to construct matching pools that produce valid offsprings. Finally, extensive experiments are carried out to demonstrate SPEA/ARP performance by comparing it with six state-of-the-art many-objective evolutionary algorithms on 5-, 10-, 15-objective of 31 benchmark MaOPs. The experiments show that SPEA/ARP outperforms the compared algorithms.

Introduction

In the real world, multi-objective optimization problems (MOPs) frequently concern several conflicting objectives. Without loss of generality, the mathematical expression for an MOP is as follows [1]:minF(x)=(f1(x),f2(x),,fm(x))Ts.t.xΩwhere x=(x1,x2,,xD)Ωdenotes the decision vector, ΩRD denotes the decision space. F:xyRm, where m denotes the number of objectives, Rm represents the objective space. When the number of objectives exceeds three (m>3), it is referred to as a many-objective optimization problem (MaOP).

Because the objectives are incompatible, there is no single solution that optimizes all of them. Instead, there is a set of tradeoff solutions known as Pareto optimal solutions [2]. The following are some basic definitions:

  • Pareto Dominance: Given two solutions xa and xb, if and only if for all dimension i,fi(xa) fi(xb) and there exists at least one dimension i,fi(xa) < fi(xb), then xa is said to dominate xb, denoted as xa xb.

  • Pareto optimal solution: If a solution x Ω satisfies condition gxΩ:xx,x is called Pareto optimal solution

  • Pareto Set: The union of all Pareto optimal solutions is called as Pareto Set (PS={x|g x Ω: x x}).

  • Pareto Front: The mapping of the Pareto Set (PS) on the objective space is called the Pareto Front (PF={F(x) Λ |x PS}).

In the last two decades, a number of multi-objective evolutionary algorithms (MOEAs) for solving MOPs and MaOPs have been successfully developed to approximate the Pareto optimal set. The existing MOEAs can be loosely categorized into four categories based on their environmental selection strategies: 1) dominance-based; 2) indicator-based; 3) decomposition-based; 4) others.

The first category is the dominance-based MOEAs, which employ Pareto dominance relationships to sort and choose potential solutions in a population before applying a second criterion to maintain population diversity. For example, NSGA-II [3] and SPEA2 [4] are two representative MOEAs based on dominance. However, as the number of objectives increases, the performance of dominance-based MOEAs is severely affected due to dominance resistance. To increase selective pressure, several dominance-based approaches have been proposed in recent years, including Cα-dominance [5], fuzzy Pareto dominance [6], and grid dominance [7]. The modified dominance criterion improves convergence performance when dealing with many-objective problems. Furthermore, dominance-based approaches frequently require another metric to balance the convergence and diversity of non-dominated solutions. For instance, Zhou et al. [8] proposed an adaptive convergence-diversity balanced evolutionary algorithm (ACDB-EA) to maintain the balance between convergence and diversity adaptively during the evolutionary process. Xue et al. [9] developed a novel population maintenance method to select high-quality solutions in the environmental selection procedure.

The second category is indicator-based MOEAs, which use performance metrics to guide the evolution of solutions. For example, S-metric selection-based MOEA (SMS-EMOA) [10] and HV contribution based on the Monte Carlo sampling algorithm (HypE) [11]. However, the computational cost of the HV indicator increases sharply with the increase of the number of objectives. Different from the HV indicator, the R2 indicator is calculated based on the reference vector and achieves a comprehensive balance of convergence and diversity. The R2 indicator-based many-objective metaheuristic II (MOMBI-II) is one example [12]. In addition, other indicators achieve success in solving many-objective problems, such as generational distance (GD) [13] and inverted generation distance (IGD) [14].

The third category of MOEAs is based on decomposition. MOEA/D [15] is a traditional decomposition-based MOEA that divides an MOP into a number of single-objective problems (SOPs) and optimizes all subproblems using a set of reference vectors. The solutions corresponding to each subproblem constitute the optimal solution set, and therefore, updating the candidate solutions using a set of predefined uniformly distributed reference vectors is gradually becoming a hot research topic. For example, Deb et al. [16] advocated decomposing the original objective space using reference vectors. Cheng et al. [17] used a collection of reference vectors and designed an angle-penalized distance to balance the convergence and variety of candidate solutions. Zhao et al. [18] proposed an automatic estimation mechanism based on the modified ant colony algorithm to assist the co-evolution between subproblems. Gu et al. [19] proposed a new reference points-based strengthened dominance relation (RPS-dominance) to increase the selection pressure. However, MOEAs based on uniform reference vectors have difficulty searching for the full solution when the true frontier is irregularly shaped, resulting in degraded algorithm performance. Therefore, many researchers propose the mechanism of adjusting vectors to solve irregular frontier problems. For example, Tian et al. [20] designed a multi-objective enhanced inverted generational distance indicator method with reference points adaptation. According to the present distribution of candidate solutions, Zhang et al. [21] developed an adaptive reference vectors technique. Zou et al. [22] presented a co-guided evolutionary algorithm that incorporates dominance and decomposition as well as reference points adaptation.

The fourth type of algorithms employs a selection mechanism that is independent of the previous three methods: 1) The fundamental method is to limit the number of objectives by eliminating redundant objectives. For example, Ma et al. [23] used principal component analysis (PCA) to do correlation analysis between objectives to reduce the dimensionality of the objective space and then used a new grid division method to ensure the diversity of populations. 2) Preference-based techniques direct algorithm updates. For example, Rivera et al. [24] introduced an ant-colony optimizer that embeds an outranking model to bear the vagueness and ill-definition of the DM’s preferences. 3) A novel category of algorithms, such as the hyperplane-assisted evolutionary algorithm (hpaEA) [25], the MaOEA with determinantal point processes (DPPs) [26], and the self-regulated bi-partitioning evolution (SBEA) [27].

According to the above description, most MOEAs have been validated against various types of benchmark MaOPs. MOEAs are primarily confronted with the enormous issues of selection pressure, diversity of solutions, and the shape of Pareto fronts while solving many-objective problems. Taking these concerns into account, we present a strength Pareto evolutionary algorithm based on adaptive reference points (SPEA/ARP). The following are the primary contributions to this paper:

  • The current and historical population information are utilized to update reference points. In this article, the active reference points are chosen based on the angle between the current population information and the preset uniform reference points. Adaptive reference points are selected from historical population information updated by shift-based density estimation and are projected into the reference plane. The adjusted reference points can effectively learn different Pareto front shapes and strengthen effective search.

  • The environmental selection process is divided into two criteria based on the fitness function values: 1) Based on the adaptive reference points, the angle distance scaling function values are used to strengthen the selection pressure. The angle-based secondary selection strategy balances the diversity of non-dominated solutions. 2) Otherwise, the fitness function values are used to select all non-dominated solutions.

  • The matching pool is constructed using an aggregate fitness r-value calculated based on the angular distance scaling function value of adaptive reference points. This method improves the adaptive search ability of the population and balances the convergence and diversity of the population.

  • Multi-objective benchmark instances encompassing various Pareto frontier problems are tested to verify the adaptability of the proposed SPEA/ARP. The experimental results show that SPEA/ARP is superior to the six state-of-the-art MOEAs.

The remainder of this work is laid out as follows. The present decomposition-based MOEAs and reference point adaptation-based MOEAs are described in Section 2. In addition, Section 2 discusses the motivation for this paper. The presented algorithm’s framework, as well as thorough descriptions of its components, are described in Section 3. The experimental settings, results, and analysis are presented in Section 4. Section 5 constructs a multi-objective optimization problem for the wet flue gas desulfurization process with seven algorithms. This paper concludes with Section 6.

Section snippets

Related work

We introduce the work relevant to this study in this section, which includes MOEAs with fixed reference points and MOEAs with adjusted reference points.

Proposed algorithm

SPEA/ARP is the presented algorithm, and the fundamental idea is to update the reference points for environmental selection using the current and archive populations rather than predefining a set of reference points that are evenly distributed. The angle distance scaling function and the angle-based secondary selection approach are constructed to improve the diversity of non-dominated solutions. Furthermore, aggregate fitness r-value is employed to construct a matching pool, enhancing the

Experimental results and analysis

In this section, we compare the proposed SPEA/ARP algorithm to six MOEAs for MaOPs with various Pareto fronts, namely MOEA/D-M2M [40], NSGAIII [16], RVEA [17], A-NSGAIII [34], MOEA/D-AWA [32], and ARMOEA [20]. MOEA/D-M2M, NSGAIII and RVEA are popular decomposition-based MOEAs that have demonstrated competitive performance on MaOPs with regular Pareto fronts; A-NSGAIII, MOEA/D-AWA and ARMOEA employ an adaptive reference points approaches to deal with irregular Pareto fronts.

In the studies, 31

Multi-objective optimization of wet flue gas desulfurization

Nowadays, multi-objective optimization algorithm research has been widely applied in many fields, for example, multi-objective reservoir flood control operation problems [47], wastewater treatment processes [48], etc. In this paper, we apply the SPEA/ARP algorithm to the optimization of a complex wet flue gas desulfurization process.

Conclusion

In this paper, we propose a strength Pareto evolutionary algorithm based on adaptive reference points, termed SPEA/ARP, for solving MaOPs with irregular Pareto fronts. Population information and historical population information work simultaneously to adjust the reference points, enabling the proposed SPEA/ARP to learn diverse Pareto front shapes more effectively and strengthen successful searches. A shift-based density estimate has been incorporated into the archive solutions so that it can be

CRediT authorship contribution statement

Xin Li: Conceptualization, Methodology, Investigation, Data curation, Writing - original draft, Writing - review & editing. Xiaoli Li: Conceptualization, Supervision, Writing - review & editing, Project administration, Funding acquisition. Kang Wang: Supervision, Writing - review & editing, Project administration. Shengxiang Yang: Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors would like to thank the editor and reviewers for their helpful comments and suggestions to improve the quality of this paper. This study was supported by the National Natural Science Foundation of China(61873006) and Beijing Natural Science Foundation(4212040, 4204087).

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