Elsevier

Swarm and Evolutionary Computation

Volume 44, February 2019, Pages 597-611
Swarm and Evolutionary Computation

User-preference based decomposition in MOEA/D without using an ideal point

https://doi.org/10.1016/j.swevo.2018.08.002Get rights and content

Abstract

This paper proposes a novel decomposition method based on user-preference and developed a variation of the decomposition based multi-objective optimization algorithm (MOEA/D) targeting only solutions in a small region of the Pareto-front defined by the preference information supplied by the decision maker (DM). This is particularly advantageous for solving multi-objective optimization problems (MOPs) with more than 3 objectives, i.e., many-objective optimization problems (MaOPs). As the number of objectives increases, the ability of an EMO algorithm to approximate the entire Pareto front (PF) is rapidly diminishing. In this paper, we first propose a novel scalarizing function making use of a series of new reference points derived from a reference point specified by the DM in the preference model. Based on this scalarizing function, we then develop a user-preference-based EMO algorithm, namely R-MOEA/D. One key merit of R-MOEA/D is that it does not rely on an estimation of the ideal point, which may impact significantly the performances of state-of-the-art decomposition based EMO algorithms. Our experimental results on multi-objective and many-objective benchmark problems have shown that R-MOEA/D provides a more direct and efficient search towards the preferred PF region, resulting in competitive performances. In an interactive setting when the DM changes the reference point during optimization, R-MOEA/D has a faster response speed and performance than the compared algorithms, showing its robustness and adaptability to changes of the preference model. Furthermore, the effectiveness of R-MOEA/D is verified on a real-world problem of reservoir flood control operations.

Introduction

In the past two decades, evolutionary multi-objective optimization (EMO) algorithms have received much attention. By evolving a population of candidate solutions with respect to conflicting objectives, EMO algorithms have the capability of generating a set of desirable non-dominated solutions in a single run. This is a significant advantage over those traditional methods from the field of Multi-Criteria Decision Making (MCDM) [1]. These days, EMO is seen increasingly by many as one of the most successful techniques in solving multi-objective optimization problems (MOPs), which involve multiple conflicting objectives to be optimized simultaneously.

Based on the different selection mechanisms, EMO algorithms usually fall into three categories, the Pareto-dominance-based algorithms [2], the indicator-based algorithms [3], and the decomposition-based algorithms [4]. Although these algorithms perform well on MOPs with two or three objectives, research works have shown that their performances suffer different degrees of deterioration as the number of objectives increases. In other words, they all show difficulties in solving many-objective optimization problems (MaOPs), where usually more than three objectives are involved. Dominance-based EMO algorithms, such as NSGA-II [2], fail to provide enough selection pressure toward the Pareto front [5] due to the dominance resistance observed in MaOPs, meaning the incomparability of solutions caused by the rapidly increased proportion of non-dominated solutions [6]. Indicator-based algorithms do not suffer from this selection pressure problem, but the cost for computing the indicator, such as the commonly-used hypervolume indicator [7], could be a major impediment [8]. The decomposition-based algorithms suffer the least from the increase of objective numbers in MaOPs [9], despite facing difficulty in configuring weighting vectors and choosing appropriate scalarizing methods [6]. Recent studies have shown that the decomposition-based algorithms, such as MOEA/D [4], are promising for solving MaOPs [10].

In MaOPs, the number of possible solutions for a good approximation of the complete Pareto Front (PF) increases exponentially with the number of objectives [11]. Since the approximation of the complete PF becomes increasingly challenging as the number of objectives increases, it would be more desirable to focus the search on a smaller and more specific PF region in the objective space, based on the preference information provided by the decision maker (DM) [12]. By considering the DM's preferences, EMO algorithms for MaOP can produce a much smaller set of solutions that are more relevant, hence more likely to be adopted by the DM and implemented in practice [13]. With the help of the preference information, an EMO algorithm can focus better its computational effort on a small and preferred PF region rather than the entire PF, saving substantial computational cost. This sort of user-preference-based EMO algorithms has been recognized as particular promising for solving MaOPs [6].

Since the decomposition-based and user-preference-based approaches are two promising methods for dealing with MaOPs, several EMO algorithms have been developed in recent years by combining both of them in order to further boost the performance [12,14]. In this sort of decomposition and preference based approach, a preferred region in the objective space is usually first specified by using preference modeling method based on one or more reference points [15], light beam search [16], and reference direction [17], etc. The preference information is then incorporated into the decomposition-based EA by employing a pre-specified set of weight vectors [13] or adjusting these weight vectors dynamically towards the preferred region [9,18]. Despite the success of incorporating the DM's preference information into the decomposition-based EMO algorithms, two major drawbacks need to be addressed:

  • The first drawback is that these algorithms are highly dependent on the ideal point. Taking the Tchebycheff scalarizing function (see equation (2)) for an example, the algorithm can only obtain solutions that are dominated by the estimated ideal point z, as shown in Fig. 1. Since there may be no apparent way to obtain the true ideal point in advance, it is typical to estimate it using the evolving population of individuals during the optimization run. The problem is that for a user-preference-based EMO algorithm, it is expected that the individuals of an EA population will converge to a preferred PF region near the reference point zr, which can be different from that of the estimated ideal point. These EA individuals can only be used to give a rough estimation of the ideal point which could be far from being accurate. In addition, this estimation could become even worse as the number of objectives increases. Unfortunately, if the estimated ideal point cannot dominate the preferred PF (as illustrated by Fig. 1), the algorithm will struggle to obtain any acceptable solutions.

  • The second drawback is that these algorithms usually guide the search towards the preferred PF region by adjusting the weight vectors of decomposed subproblems. The adaptation of weight vectors changes the set of decomposed subproblems over the course of optimization, which may slow down the convergence speed of the optimization algorithm [19].

To overcome these problems, this paper proposes a user-preference-based EMO algorithm which does not rely on an estimation of the ideal point, and neither an adaptation of the weight vectors. Unlike existing scalarizing functions which are based on the estimated ideal point and a set of weight vectors, the proposed scalarizing function generates the decomposed subproblems by using a set of new reference points derived from the reference point specified by the DM in the preference model, and a set of unique weight vectors. Consequently, there is no need to tune a set of weight vectors in order to guide the search moving towards the preferred PF region. The main contributions of this work can be summarized as follows:

  • a)

    A new scalarizing function making use of a series of new reference points derived from a given reference point in the preference model, instead of the ideal point, is proposed.

  • b)

    Based on this scalarizing function, a user-preference-based EMO algorithm, namely R-MOEA/D, is developed. The key merit of R-MOEA/D is that it neither rely on an estimation of the ideal point nor on an adaptation of the weight vectors.

  • c)

    The proposed scalarizing function can be conveniently plugged into any decomposition-based EMO algorithm to obtain solutions covering the preferred PF region rather than the whole PF.

The remaining of this paper is organized as follows. Section 2 provides the background information and related work. Section 3 describes the proposed reference point based scalarizing function and the new R-MOEA/D algorithm. Section 4 demonstrates the superiority of R-MOEA/D through a series of experimental studies. Section 5 verifies the effectiveness of R-MOEA/D on real-world reservoir flood control operation problems. Finally, concluding remarks are given in section 6.

Section snippets

Backgrounds and related work

This section presents the background information and related work, including the definition of MOPs and MaOPs, the decomposition-based EMO algorithm MOEA/D, EMO algorithms combining decomposition and user preferences, and the reference point based preference model used in this work.

The reference point guided MOEA/D

In this section, a reference point based scalarizing function which is independent from the ideal point is first proposed. With this new scalarizing function, a decomposition-based EMO algorithm can be easily converted into a user-preference-based EMO algorithm without changing any of its process. To demonstrate its usefulness, we embed the new scalarizing function into the algorithmic framework of MOEA/D, giving rise to the proposed R-MOEA/D algorithm.

Experimental studies

In this section, R-MOEA/D is compared with two other reference point based EMO algorithms R-NSGA-II [15] and R-MEAD2 [12]. Experiments are conducted on multi-objective and many-objective benchmark functions including ZDT [47], DTLZ [48] and WFG [49] test suites, to demonstrate the superiority of the proposed algorithm.

The proposed R-MOEA/D algorithm was implemented in C++ programming language using Visual Studio 2013. The source codes of the compared algorithms were download from the home page

Performance comparisons on RFCO problems

The Reservoir flood control operation (RFCO) problem is a challenging real-world problem with an aim to reduce flood peaks and damages simultaneously during floods by making appropriate scheduling plans on a dam's water release sequences [[54], [55], [56]]. To guarantee the safety of the upstream side of the dam, the highest upstream water level should be minimized. As for the safety of the downstream side, the largest discharging water flow must not be too large. Given a flood inflow sequence,

Conclusion

User-preference-based and decomposition-based evolutionary multi-objective optimization (EMO) algorithms are two promising approaches for dealing with many-objective optimization problems (MaOPs). In recent years, several EMO algorithms have been developed by combining these two approaches. One common deficiency of this class of EMO algorithms is that they often need to make use of an ideal point estimated from the evolving population. However, this estimation of the ideal point is often

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No. 61303119, the Science Basic Research Plan in Shaanxi Province of China under Grant No. 2018JM6009, the Fund Program for the Scientific Activities of Selected Returned Overseas Professionals in Shaanxi Province of China under Grant No. 2017021, and the Fundamental Research Funds for the Central Universities under Grant No. JB140304. The first author would also like to thank the ECML research group at RMIT

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