# Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
## 1 Introduction  
In the real world, many problems can be described as multi-objective optimization problems (MOPs), where multiple objectives often conflict with each other. Each objective has to compromise in optimization, and finally generate a set of balanced solutions, called the Pareto optimal set (POS) [1], which is provided to decision makers. Evolutionary multi-objective optimization, referred to EMO, focuses on using evolutionary algorithms for MOPs [2]. Its research has become a hot topic in the field of evolutionary computation. MOPs can be further divided into static and dynamic multi-objective problems (DMOPs). In the dynamic environment, the objective function, constraint function, and the related parameters of DMOPs can change with time [3]. Therefore, a dynamic evolutionary multi-objective optimization algorithm (DEMOA) must be able to automatically detect and respond to new changes with fast convergence, and track the time-varying POS in a timely fashion. As a result, this study brings new challenges to EMO [4]. In the past ten years, some researchers have had strong interest in dynamic EMO (DEMO) [5; 6]. Based on genetic algorithms, artificial immune algorithms [7; 8], particle swarm optimization algorithms [9; 10], co-evolutionary algorithms [11], membrane computing [12; 13], and other natural computing methods, they designed the corresponding DEMOAs [14]. DEMOAs have been preliminarily used in intelligent service [15], industrial design [16],engineering management [17], scheduling control and optimization [18], and other fields. For their further researches have important theoretical significance and practical applied value. How to make a rapid response to a new environment using an obtained optimal solution is very important in the design of a DEMOA. Memory methods can reuse the previous optimal solutions by remembering to help the algorithm make a good response to the new changes. There are some dynamic single objective evolutionary algorithms with memory methods to enhance the dynamic tracking performance [19; 20]. So far, however, few memory methods are used in DEMOAs. In 2007, the famous dynamic non-dominated sorting genetic algorithm II (DNSGA-II) [16] was proposed by [PERSON] et al., which was based on NSGA-II [21]. In 2009, [PERSON] et al. proposed a collaborative multi-objective evolutionary algorithm for dynamic competition cooperation, named the dynamic competition-cooperation co-evolutionary algorithm (dCOEA) [12]. [PERSON] and [PERSON] et al. proposed a dynamic multi-objective evolutionary gradient search algorithm (dMO-EGS) [22] in 2010. [PERSON] et al. proposed an immune clonal coevolutionary algorithm for dynamic multi-objective optimization (QICCA) [7] in 2014. All these DEMOAs have tried to use memory methods to quickly respond to new changes, but the actual effects are not ideal. In addition to rapid response and dynamic tracking, another difficulty in designing a DEMOA is that the Pareto solutions set obtained by the algorithm should has good convergence and distribution in each process of environment change. At present, most DEMOAs follow the traditional approaches of static evolutionary multi-objective optimization and use the Pareto domination and elite reserved strategies to ensure the convergence. They also use the operators of diversity maintaining to ensure the diversity of the Pareto solutions. [PERSON] et al. proposed a new framework called the multi-objective evolutionary algorithm based on decomposition (MOEA/D) for solving static MOPs in 2007 [23]. MOEA/D decomposes a multi-objective problem into a number of single-objective sub-problems and solves these subproblems in parallel. MOEA/D has become very popular in solving static MOPs, but fewer decomposition-based DMOEAs are researched for DMOPs. In this paper, we introduce this framework and propose a memory-enhanced dynamic multi-objective algorithm based on \(L_{p}\) decomposition for solving DMOPs, where the \(L_{p}\) decomposition method is presented. The proposed algorithm is abbreviated as dMOEA/D-\(L_{p}\).