# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 2 Basic Concepts of MOO Problem and Related Research on MOEAs
### Research on Multi-Objective Evolutionary Algorithms  
As early as 1967, [PERSON] mentioned in his doctoral dissertation that genetic search algorithms can solve MOO problems but it was not until 1985 that the first MOEA Vector Evaluated Genetic Algorithm (VEGA) emerged [26]. What really attracts the evolutionary computation world is the series of MOEAs that have been proposed since 1990. The development of MOEAs can be divided into two stages [27]. The first phase was primarily composed of Pareto-based optimization and non-Pareto-based optimization. One of the typical algorithms not based on Pareto dominance was VEGA, which partitioned the population of size \(P\) to \(r\) sub-populations and each sub-population evolved for different sub-objectives respectively. The superiority of VEGA was that the operation was simple and could be implemented easily, but it was hard to obtain the optimal solution set when the true Pareto Front was non-convex. The other branch was the Pareto-based approach, which was first put forward by [PERSON] [28] in 1989 and introduced the Pareto Rank mechanism to implement the operation of selection. The Pareto-based algorithms mainly include Niched-Pareto Genetic Algorithm (NPGA) [29] proposed by [PERSON] and [PERSON], Multi-objective Genetic Algorithm (MOGA) [30] proposed by [PERSON] and [PERSON], and so forth. However, the Pareto-based approach is not universal and it is necessary to choose a method to maintain the diversity of the solution set based on specific optimization problems. The concept of \"external set\" was put forward at the second stage. The MOEAs proposed in this period emphasize efficiency and effectiveness. Among them, the typical ones are given as follows. [PERSON] and [PERSON] [27] proposed the Pareto Archived Evolution Strategy (PAES) series in 1999, [PERSON] et al. [10] brought forward the Pareto Envelope-based Selection Algorithm (PESA) for MOO problems and proposed its improved version PESA-II [11] in 2001 and [PERSON] et al. [5] put forward a non-dominated set sorting method named Non-dominated Sorting Genetic Algorithm (NSGA-II), which compensates the shortcomings of its previous version NSGA [4]. In NSGA-II, in order to reduce the time complexity of the algorithm, a fast sorting method was applied to construct a non-dominated solution set, but it was difficult to find outliers, and may generate search offset when increasing the number of objective functions [25]. To some extent, the classic MOEAs are very mature but the field of evolutionary algorithms is still evolving. In recent years, a large number of split-new MOEAs has appeared. For example, [PERSON] and [PERSON] [31] propose a fast Approximation-Guided Evolutionary multi-objective algorithm (AGE-II), which approximate the archive in order to control its size and influence on the runtime, and avoid the shortcomings of Approximation-Guided Evolution (AGE). [PERSON] and [PERSON] [15] propose the Non-dominated Sorting and Local Search (NSLS) optimization algorithm, which have better diversity in terms of combining the farthest-candidate approach with the non-dominated sorting method to select the new population members. But it easily falls into local optimum on some  
Figure 1: Pareto optimal solutions (a,b,c,d,e,f) and dominated solutions (g,h,i,j,k,l). test problems. [PERSON] et al. put forward the Bi-criterion evolution for IBEA (BCE-IBEA) ([17; 32]), it can freely implement the NPC evolution part, which made the approach effectively for those non-Pareto-based algorithm. [PERSON] et al. [18] propose a Multi-Objective Evolutionary Algorithm based on an enhanced Inverted Generational Distance metric (MOEA/IGD-NS), which improves the method of environmental selection. MOEA/IGD-NS uses an enhanced inverted generational distance metric to find non-contributing solutions (termed IGD-NS). In recent years, more and more researchers are turning their attention to many-objective optimization (MaOO) problems. For example, [PERSON] et al.