Cooperative co-evolutionary algorithm for multi-objective optimization problems with changing decision variables

Highlights

• Presenting an approach to dynamically adjusting the grouping of decision variables when the number of decision variables changes.

• Providing a hybrid strategy of initializing subpopulations to rapidly respond to the change of an optimization problem.

• Giving a distance-based strategy of selecting representatives from the other subpopulations when evaluating an individual.

Abstract

Multi-objective optimization problems (MOPs) with changing decision variables exist in the actual industrial production and daily life, which have changing Pareto sets and complex relations among decision variables and are difficult to solve. In this study, we present a cooperative co-evolutionary algorithm by dynamically grouping decision variables to effectively tackle MOPs with changing decision variables. In the presented algorithm, decision variables are grouped into a series of groups using maximum entropic epistasis (MEE) at first, with decision variables in different groups owning a weak dependency. Subsequently, a sub-population is generated to solve decision variables in each group with an existing multi-objective evolutionary algorithm (MOEA). Further, a complete solution including all the decision variables is achieved through the cooperation among sub-populations. Finally, when a decision variable is added or deleted from the existing problem, the grouping of decision variables is dynamically adjusted based on the correlation between the changed decision variable and existing groups. To verify the performance of the developed method, the presented method is compared with five popular methods by tackling eight benchmark optimization problems. The experimental results reveal that the presented method is superior in terms of diversity, convergence, and spread of solutions on most benchmark optimization problems.

Introduction

MOPs involve simultaneous optimization of two or more conflicting objectives [1], [2]. In particular, an MOP with an increasing or decreasing dimension of decision variables over time is one with changing decision variables. A multi-period portfolio selection problem, for example, mainly addresses how to rationally allocate limited funds to various financial assets to balance the maximal return and minimal risk in a number of consecutive periods [3], [4]. Owing to the changing market environment, the types and proportions of invested securities in each period are adjusted according to the current market environment. In the above process, investors are likely to hold on some of their existing assets while selling bad assets and/or buying new ones with higher returns. If assets to be sold/bought are regarded as the decision variables when formulating the portfolio selection problem, it becomes a problem with changing decision variables. Specifically, it is an MOP with changing decision variables.

The changing of decision variables will result in a changing Pareto-optimal set (PS) for an optimization problem. Therefore, an MOP with changing decision variables can be regarded as a type of dynamic multi-objective optimization problem (DMOP) [2], [5]. An approach of effectually tackling a DMOP needs to conquer difficulties raised by changing conditions within an optimization problem, such as following up a time-dependent Pareto front (PF) and offering solutions with good diversity. To this end, Deb et al. introduced diversity by randomly initializing a population or conducting the mutation operator on certain solutions chosen from the population [6]; Liang et al. incorporated a hybrid of memory and prediction strategies into a multi-objective evolutionary algorithm based on decomposition (MOEA/D) [7], in which a differential prediction is employed to relocate the population individuals in the new environment if a detected change is dissimilar to any historical changes, and a memory-based technique devised to predict the new locations of the population members is applied if a similar environment exists in the historical changes. In [8], a dynamic multi-objective evolutionary algorithm driven by inverse reinforcement learning is used to tackle the DMOPs. To accelerate the convergence and maintain the diversity of the evolutionary population, a Q-learning-based change response approach is considered to generate solutions in the promising regions; and [9] presented a multidirectional prediction strategy by clustering a population into several representative groups to improve the performance of evolutionary algorithms (EAs) in solving a DMOP. Recently, Li et al. provided a dynamic two-archive EA to tackle DMOPs with a changing number of objectives [10]. Two complementary co-evolving populations are simultaneously maintained to adaptively reconstruct their compositions once the environment changes and interact with each other via a mating selection mechanism.

However, all these methods have not considered the case where the number of decision variables increases or decreases over time. Although studies have been conducted on changing the number of decision variables for single-objective problems, such as [11], [12], they consider different problems from those in our study. They aim to provide an approach for determining an optimal search dimension with a small population size when tackling a problem that has an infinite search dimension. To this end, an evolutionary algorithm is first employed to address an optimization problem with a small number of decision variables. Then, the number of decision variables is gradually increased during the optimization. The increase in search dimension is continued until performance cannot be improved within a certain number of generations after the dimension increases. In this case, the search dimension is decreased by one and remains unchanged. Finally, an optimal search dimension for a small population is provided. This indicates that the increasing or decreasing number of decision variables is only considered a strategy for solving the problem, which is not the characteristic of the problem. For a large-scale MOP with changing decision variables, running the current optimization method based on a new population is inefficient when a change in the number of decision variables is detected. Furthermore, producing a complete set of solutions to a DMOP with a large scale of decision variables using previous methods often results in an insurmountable computational complexity, suggesting their inefficiency in tackling an MOP with changing decision variables. Therefore, seeking appropriate methods for an MOP with changing decision variables is critical, and it is the key focus of this study.

It has been shown that co-evolutionary mechanisms can significantlyimprove the efficiency of the optimization process [2], [13]. Because cooperative co-evolutionary algorithms (CCEAs) can significantly shrink the searching space of a sub-population, they are efficient when solving a single-objective large-scale optimization problem [14], [15], [16]. In addition, CCEAs have been employed in conjunction with other strategies to address MOPs and DMOPs. For example, Li et al. developed a systematic way of incorporating the decision maker’s preference information into the decomposition-based evolutionary multiobjective optimization methods [17]. Therefore, the search process is steered toward the region of interest for the decision maker directly or interactively. Moreover, to help decision makers identify solution(s) of interest from a given set of trade-off solutions in an MOP, they presented a simple and effective knee point identification method from a decomposition perspective [18]. The basic idea is to sequentially validate whether a solution is a knee point or not by comparing its localized trade-off utility with others within its neighborhood characterized from a decomposition perspective. However, they decomposed the problems in the objective space, which is difficult to adapt to problems with changing decision variables. [19] presented a distributed CCEA by exploiting the inherent parallelism of cooperative co-evolution (CC), which divides an MOP into several sub-problems based on the decision variables; each sub-problem contains only one decision variable and is optimized by a sub-population. [20] proposed a dynamic competitive-cooperative co-evolutionary algorithm to address DMOPs, where all the decision variables are adaptively classified into several groups and random competitors are utilized to track the moving optima. Two approaches to large-scale multi-objective optimization were introduced in [21], [22]. One is to solve an MOP with many decision variables, called an evolutionary algorithm for large-scale many-objective optimization (LMEA), which divides the decision variables into distance- and diversity-related groups using a clustering approach [21]. The other is an MOEA based on decision variable analyses (MOEA/DVAs) for large-scale MOPs [22], which groups the decision variables according to the contribution of a decision variable to convergence (i.e., the distance to the PF), diversity, or both. However, many function evaluations are consumed before the optimization, especially for an optimization problem with numerous decision variables. Furthermore, these studies can only provide the correlation between decision variables, and are mainly suitable for decomposing decision variables of an optimization problem with separable objectives.

In recent years, researchers have proposed several novel grouping methods, such as differential grouping (DG), an improved variant of DG (DG2), global DG (GDG), and recursive DG (RDG). The DG [23] method identifies the interaction between decision variables by detecting the fitness changes when perturbing the decision variables. If the fitness change induced by perturbing decision variable $x_i$ varies for different value of $x_j,x_i$ and $x_j$ interact. However, DG is sensitive to the value of $\epsilon$ which is a parameter of DG used for determining whether two variables are nonseparable. To overcome this defect, DG2 [24] is proposed to adapt the value of $\epsilon$ to the objective value of a problem and identify the complete variable interaction matrix. Thus, the accuracy in identifying the interrelationship is improved. However, it may not need the entire variable interaction matrix to identify the connected subcomponents. In theGDG [25] method, the same technique as that used in DG is employed to identify the pairwise interactions between decision variables. The variable interaction matrix, also know as the adjacency matrix of a graph, is calculated. Then, the depth-first search or breadth-first search is used to identify the connected components. Note that both the interacting and conditionally interacting decision variables are placed into one connected subcomponent. The computational cost of DG, DG2, and GDG for decomposing a D-dimensional problemis $O(D^2)$. To reduce the computational cost of the differential grouping methods, RDG examines the interrelationship between a pair of sets of variables but not a pair of variables [26], [27], [28]. If two sets of variables ($X_1$ and $X_2$) are interrelated, RDG divides $X_2$ into two equal-sized subsets and examines the interrelationship between $X_1$ and the two subsets. The computational complexity is $O({\mathit{Dlog}}_{2}D)$ when RDG decomposes a D-dimensional problem in the above binary search fashion. Nevertheless, the mentioned grouping methods, DG, DG2, GDG, and RDG, which divide the decision variables offline, can not adapt to problems associatedwith decision variable change. They lack a mechanism for adjusting the groups of decision variables as the optimization problem varies.

Therefore, it is extremely challenging for CCEAs to tackle an MOP with changing decision variables. The challenges include the following: (1) how to examine the change in an environment, (2) how to adjust the groups of decision variables when the environmental change occurs, and (3) how to respond to the environmental change. Bearing these challenges in mind, we provided some preliminary results [29]. In [29], a framework of parallel cooperative co-evolution based on dynamically grouping decision variables is proposed. With the Spearman rank correlation (SRC) analysis based on samples obtained from the evolution of a population, the decision variables are partitioned into several groups, which dynamically adjust when a change occurs. However, the samples have a significant influence on the above grouping results, causing inaccurate grouping results and low reliability. Moreover, only the case of increasing decision variables is considered, which is not adequate.

In this study, we further extend the previous work. An MOP with one decision variable being increased or decreased at a time is considered. Note that if more than one decision variable is changed simultaneously, we can handle them individually. A CCEA is presented for tackling this optimization problem. In this study, a number of groups are gained at first based on the relation among decision variables. As the dimension of decision variables increased or decreases, the interaction matrix (IM) between a newly added decision variable or an old reduced decision variable and each group will be computed according to information offered by the population to adjust the grouping of decision variables. Additionally, a hybrid strategy is employed to initialize sub-populations as the dimension of decision variables varies.

Some new features different from the previous work are provided as follows.

• (1) Employing a novel method to accurately group the decision variables.

• (2) Proposing an improved strategy to respond the change of decision variables.

• (3) Detailing the strategy of evaluating an individual of a sub-population.

• (4) Employing four new metrics to reflect the performances of the presented algorithm, and extending the experiments to survey the influences of different strategies on the proposed algorithm, which are beneficial to enriching the experiments.

The remainder of this paper is structured as follows. A comprehensively review on the related work is provided in Section 2. Section 3 details the proposed CCEA based on dynamically grouping decision variables. The experimental results are reported and analyzed in Section 4. Finally, Section 6 concludes the whole study.