**1. Introduction**

In recent years, with the development of technology, more and more new problems appear in the field of industry or control and so on. This problem is usually characterized by containing more than one objective function to be optimized, and these objective functions contradict each other. Such problems are generally called multi-objective optimization problems (MOPs) or many-objective optimization problems (MaOPs). Generally, MOPs are problems having two or three objectives, and MaOPs contain more than four objectives [1]. A MaOP is defined as follows:

$$\begin{array}{ll}\text{minimize} & F(\mathbf{x}) = \{f\_1(\mathbf{x}), f\_2(\mathbf{x}), \dots, f\_M(\mathbf{x})\} \\ & \text{subject to} \quad \mathbf{x} \in X \end{array} \tag{1}$$

where *M* is the objective number. *x* = (*<sup>x</sup>*1, *x*2, ... , *xn*) is decision variable, and *X* ⊆ *Rn* is the decision space of the *n*-dimensional real number field. *F* is a mapping from decision space to objective space, and *F*(x) contains *M* different objective functions *fi*(*x*) (*i* = 1, 2, . . . , *M*).

**Citation:** Li, G.; Wang, G.-G.; Wang, S. Two-Population Coevolutionary Algorithm with Dynamic Learning Strategy for Many-Objective Optimization. *Mathematics* **2021**, *9*, 420. https://doi.org/10.3390/ math9040420

Academic Editor: Alicia Cordero

Received: 1 February 2021 Accepted: 19 February 2021 Published: 20 February 2021

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Generally, the optimal solutions of MaOPs are distributed on Pareto front (PF), and the solutions on PF generally show the trade-off on all *M* objectives. Therefore, it is impossible to obtain an optimal solution by an optimization method to make it optimal on all objectives. What is needed to solve MaOPs is to obtain a set that has a finite number of solutions, so that the solutions in this solution set can well represent the whole PF, no matter in terms of convergence or diversity. The EAs had its unique advantages in solving MaOPs because of its population-based characteristics. After years of development, scholars from all over the world have proposed various many-objective optimization algorithms (MaOEAs) to solve MaOPs. Because different MaOPs also have different characteristics, currently no general algorithm can perfectly solve all MaOPs.

MaOPs usually have more than three objective numbers, so the objective space cannot be visualized, and the high-dimension calculation equation can only be obtained through derivation in solutions with fewer objectives. For example, grid-based evolutionary algorithm (GrEA) [2] deduces the high-dimensional grid calculation equation through a two-dimensional equation. Moreover, with the increasing of the objective number, the number of non-dominated individuals in the population will also increase exponentially. Studies have shown that almost all the solutions in the obtained population will be nondominated when *M* > 12 [3]. As a result, the selection pressure of the algorithm based on non-dominated sorting is reduced, which makes the algorithm unable to solve the MaOPs well. In addition, the shape and density of PF vary greatly for different problems, which brings grea<sup>t</sup> challenges to obtaining a solution set with good convergence and diversity.

To overcome these difficulties with MaOPs, a number of MaOEAs have been proposed. For example, on the basis of fast and elitist multi-objective genetic algorithm (NAGA-II) [4], reference points are introduced to guide individual convergence and help evolution through the concept of domination, called NSGA-III [5,6]. Knee point-driven evolutionary algorithm (KnEA) [7] used knee points to guide individual convergence, and GrEA introduced the concept of grid to choose the better of two non-dominated individuals. In addition, some indicator-based algorithms are proposed, such as indicatorbased selection in multi-objective search (IBEA) [8] and fast hypervolume-based algorithm (HypE) [9], which adopt *I*ε+ [10] and hypervolume (HV) [11,12] indicators as the criteria for selecting individuals, respectively. The selection process is the evolution process of the whole population or individuals towards the direction with better indicator values. Finally, there is the evolutionary algorithm based on decomposition (MOEA/D) [13], which adopts the idea of mathematical decomposition to decompose a MaOP into *M* subproblems for simultaneous optimization. Common decomposition approaches include weighted sum approach, Tchebycheff approach, and penalty-based boundary intersection approach. There is a new way called MOEA/D-PaS [14] to combine the decomposition with Pareto adaptive scalarizing methods to balance the selection pressure toward the Pareto optimal and the algorithm robustness to PF.

At the same time, these MaOEAs also contain some disadvantages. The convergence speed of the Pareto-based algorithm is slow, or even unable to converge to PF. Indicatorbased algorithms often tend to favor one or some of special regions of PF. Although the convergence speed of an indicator-based algorithm is generally relatively fast, the diversity of the solution set is usually poor. Besides, the decomposition-based MaOEAs are very dependent on the selection of decomposition approaches, such as weighted sum method in dealing with the non-convex problem (in the case of minimization) of PF shape, and not all Pareto optimal vectors can surely be obtained [13]. In addition, when the shape and density of PF are very complex and changeable, the traditional method to generate a weight vector is not suitable for this environment.

Existing multi-objective optimization algorithms (MOEAs) will still be affected by the increased number of objectives when dealing with MOPs, especially MAOPs. Moreover, the complexity of the PF also brings grea<sup>t</sup> challenges to the MOEAs. However, the dynamic learning strategy can pay more attention to the improvement of convergence when the population has poor convergence in the early evolutionary stage, and pay more attention

to the improvement of diversity in the late evolutionary stage. In addition, coevolution is a promising idea to improve the quality of individuals in a population through the mode of cooperation (or competition) between multiple populations (or subpopulations). Through coevolution, some key information of the population (such as the evolutionary state of the population) can be obtained in the process of algorithm iteration. The information of the population can be fed back flexibly to change the dynamic level (this will be discussed in Section 4) of dynamic learning strategies. In conclusion, the combination of a dynamic learning strategy and a coevolution model is a promising way to improve the convergence and diversity of the population.

The rest of this paper will be arranged as follows. Section 2 introduces the related works of the coevolutionary algorithm, other background, and the motivation for a twopopulation coevolution algorithm with dynamic learning strategy (DL-TPCEA). Section 3 will give the background of the MaOPs. The algorithm framework, process details, and parameter settings will be introduced in Section 4. Sections 5 and 6 carry out the analysis of experiments and the conclusion, respectively.
