Complexity

Volume 2019, Article ID 3251349, 11 pages

https://doi.org/10.1155/2019/3251349

## A Constrained Solution Update Strategy for Multiobjective Evolutionary Algorithm Based on Decomposition

College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China

Correspondence should be addressed to Qiuzhen Lin; nc.ude.uzs@nilhzuiq

Received 28 November 2018; Accepted 23 January 2019; Published 8 May 2019

Academic Editor: Alex Alexandridis

Copyright © 2019 Yuchao Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a constrained solution update strategy for multiobjective evolutionary algorithm based on decomposition, in which each agent aims to optimize one decomposed subproblem. Different from the existing approaches that assign one solution to each agent, our approach allocates the closest solutions to each agent and thus the number of solutions in an agent may be zero and no less than one. Regarding the agent with no solution, it will be assigned one solution in priority, once offspring are generated closest to its subproblem. To keep the same population size, the agent with the largest number of solutions will remove one solution showing the worst convergence. This improves diversity for one agent, while the convergence of other agents is not lowered. On the agent with no less than one solution, offspring assigned to this agent are only allowed to update its original solutions. Thus, the convergence of this agent is enhanced, while the diversity of other agents will not be affected. After a period of evolution, our approach may gradually reach a stable status for solution assignment; i.e., each agent is only assigned with one solution. When compared to six competitive multiobjective evolutionary algorithms with different population selection or update strategies, the experiments validated the advantages of our approach on tackling two sets of test problems.

#### 1. Introduction

In real-world applications, it is often needed to handle multiobjective optimization problems (MOPs) [1], such as recommendation systems [2, 3], privacy computing [4], and resource assignment [5–7]. Due to the conflicts among different objectives, the results of MOPs will output a set of Pareto solutions (PS) and their mapping in the objective space is called Pareto front (PF) [8–10]. These MOPs may be characterized with complicated features [11–13], which cannot be well solved by traditional mathematical methods. Instead, multiobjective evolutionary algorithms (MOEAs) can effectively obtain a set of solutions in one single run, which have shown a very promising performance in tackling different kinds of MOPs [14–16] and become very popular during the recent decades.

In the design of MOEAs, evolution and selection are their two important mechanisms [17–19]. The first one modifies individuals in order to approach the true PF, while the second one selects the most promising individuals to constitute the new population for next generation. Based on the selection mechanisms, most MOEAs can be classified into three types, i.e., Pareto-based MOEAs [20–23], indicator-based MOEAs [24–29], and decomposition-based MOEAs [30–36]. Compared to the selection operators used in Pareto-based and indicator-based MOEAs, decomposition-based MOEAs are able to provide more flexibility to balance convergence and diversity [37], which has been found to provide a better performance when tackling some complicated MOPs, as reported in [38]. In this sort of MOEAs, the target MOP is decomposed into a set of subproblems, which are solved simultaneously using a set of cooperative agents. Each agent aims to optimize one subproblem in MOEA/D [39]. Due to the simplicity and effectiveness of MOEA/D, this framework has triggered a considerable amount of research, aiming to improve different components of MOEA/D, such as the adjustment and generation of weight vectors [40–44], dynamic resource allocation [45–47], enhanced evolutionary operators [48–50], and improved population selection or update mechanisms [51–57].

Especially, regarding the population selection or update mechanisms for decomposition-based MOEAs, the offspring in MOEA/D [39] are allowed to update any solution in population. However, this method may significantly lower the diversity when a very good solution may replace most of the others in several generations. In MOEA/D-DE [58], its solution update approach is controlled by two preset probabilities and* n*_{r}, which obtains a better balance of convergence and diversity. The offspring is only allowed to update the parent solutions from the neighborhood with a probability and from the entire population with a probability (1-). Moreover, an offspring can only replace at most* n*_{r} parent solutions. This strategy was mostly used in the following design of decomposition-based MOEAs [49, 54]. Different from the decomposition approach in MOEA/D-DE, MOEA/D-M2M [38] separates the search space into multiple search subspace, which simples the solving of MOPs in each subspace and the solution update is constrained by including the equal number of solutions in each subspace. Thus, MOEA/D-M2M was shown to be very effective for complicated MOPs that strongly emphasize diversity (i.e.,* MOP* problems [38]). To further find a better match of solutions and subproblems, a stable matching model was proposed in MOEA/D-STM [59], which associates the solutions to subproblems according to their respective preferences. In this way, MOEA/D-STM can maintain a good convergence speed and population diversity. Similarly, an improved interrelationship model was designed in MOEA/D-IR [37] to associate the solutions to subproblems based on their mutual-preferences, which is an essentially diversity first and convergence second strategy. Moreover, two improved versions [53] for MOEA/D-STM were proposed to embed the concept of the incomplete preference lists in the stable matching model, which further strengthens the diversity. In [51], an adaptive replacement neighborhood size was proposed to assign an offspring to its most appropriate subproblems, obtaining a better balance of convergence and diversity. In MOEA/D-ACD [54], an adaptive constrained decomposition approach was presented, in which the update regions of decomposition approach are constrained to maintain the diversity. Moreover, to further enhance the performance in MOPs with more than three objectives, decomposition approach and Pareto domination were simultaneously used in MOEA/DD [44], decomposition-based-sorting and angle-based-selection approaches were proposed in MOEA/D-SAS [57], and the diversity was preferred in solution update by selecting certain closest subproblems for an offspring in [60].

On the other hand, another kind of population selection or update mechanisms in MOEA/D aims to improve their used decomposition functions. In MOEA/D [39], three traditional decomposition functions, i.e., the weighted sum (WS) approach, the Tchebycheff (TCH) approach, and the penalty-based boundary intersection (PBI) approach, were employed. In [61, 62], a local PBI and WS were, respectively, designed to constrain the update regions of decomposition approaches, which avoid the diversity loss. In [63, 64], an adaptive Pareto front scalarizing (PaS) and penalty-based boundary intersection (PaP) decomposition approaches were, respectively, introduced to match the true PFs with various shapes. Two decomposition approaches were presented in MOEA/AD [65] and DECAL [66] to deal with the complicated PF. In MOEA/AD, two coevolved populations were, respectively, updated by the two decomposition functions to fit different PF shapes, while two novel decomposition functions were, respectively, used to accelerate the convergence speed and enhance the population diversity in DECAL. Recently, MOEA/D-LTD [67] was proposed to trace the PF shape, in which the learning module predicts the PF shape and the decomposition function is adaptively adjusted to fit its PF shape.

Most of the above MOEAs all abide one basic principle that each agent should be assigned with one solution in order to find the optimal value for its subproblem. However, this kind of solution assignment may not be effective and efficient in decomposition-based MOEAs, as the solution assigned to the agent may be far away from its subproblem. In such case, it cannot truly reflect the diversity of each agent and cannot provide the correct neighboring information in evolution, which may slow down the convergence as decomposition-based MOEAs are designed as an essentially collaborative evolutionary framework. Therefore, a constrained solution update (CSU) strategy is designed in this paper for decomposition-based MOEAs to alleviate the above problem. The solutions are only assigned to the agent that handles the closest subproblem. This way, the correct neighboring information can be provided to guide the evolution and it is straightforward to show the diversity of each agent. In this case, the number of solutions in each agent may be zero or no less than one. To maintain the diversity of each agent, the offspring assigned to one agent are only allowed to renew its original solutions. When the agent has no solution, it will be assigned one solution in priority, once offspring are generated closest to its subproblem. To keep the same population size, the agent with the largest number of solutions will remove one solution showing the worst convergence. Thus, the diversity of one agent is enhanced, while the convergence of other agents is not affected. After a period of evolution, a stable status for solution assignment is anticipated so that each agent only has one solution. When compared to the existing population selection or update strategies for decomposition-based MOEAs, our experiments validate the superiority of the proposed approach when tackling two sets of complicated test MOPs.

The main contributions of this paper are clarified below.(1)Each agent may be assigned with no solution, or no less than one solution, which is different from the existing approaches that only assign one solution to each agent. This approach can truly reflect the diversity on the agents and provide the correct neighboring information in evolution.(2)A CSU strategy is designed for each agent in order to maintain diversity for all the agents without affecting their convergence. The agent with no solution will be assigned first, while the agent with the largest number of solutions will remove one solution showing the worst convergence. By this way, a stable status for solution assignment may be reached, so that each agent only has one solution, which ensures diversity in decomposition-based MOEAs.(3)When solution assignment is under an unstable status such that at least one agent is still not assigned any solution, the mating parents are randomly selected from the best solutions from all the agents, as the neighboring agent may have no solution. This random selection of mating parents helps to enhance the exploration ability in our algorithm.

The rest of this paper is organized as follows. Section 2 provides the related background, such as MOPs and the used decomposition function in this paper. Section 3 introduces the details of the proposed algorithm MOEA/D-CSU. The experimental results and discussions are provided in Section 4, while the conclusions and some future research directions are given in Section 5.

#### 2. Related Background

##### 2.1. Multiobjective Optimization Problems

Multiobjective optimization problems often need to optimize several conflicting objectives, which can be modeled bywhere is an* n* dimensional decision vector in the decision space Ω and* m* is the number of objectives. The target of MOP in (1) is to minimize all the objectives simultaneously.

##### 2.2. The Decomposition Function

In this paper, the modified Tchebycheff method [55] is used for decomposing the MOP in (1), which is defined bywhere is a preset weight vector with for each and , while is the ideal point by setting for each . When using* N* uniformly distributed weight vectors in (2), the MOP in (1) is decomposed into a set of* N* subproblems, which can be solved by a set of* N* collaborative agents. The population selection or update strategies designed in decomposition-based MOEAs will reasonably allocate the solutions to the agents [39]. Different from the existing approaches [39, 58] that assign one solution to each agent, the agent in our approach is only allocated by the solutions that are closest to its subproblem, resulting in the fact that the number of solutions in each agent may be zero or no less than one. To show (2) more visually, a case of updating solution is depicted in Figure 1, where* s*_{1} is a solution in current population while* s*_{2} and* s*_{3} are two offspring. For this case,* s*_{3} can update the subproblem but* s*_{2} cannot do this, because the yellow region is the improvement domain of* s*_{1} by the weight vector and (2), and a solution like* s*_{3} falling into the region can update the subproblem. Actually, (2) decides the profile of the region [54].