Multi-objective equilibrium optimizer slime algorithm (MOEOSMA) solves CEC2020 multi-mode multi-objective optimization

       The multi-objective equilibrium optimizer slime algorithm (MOEOSMA) has better optimization performance than existing multi-objective slime algorithms. In MOEOSMA, dynamic coefficients are used to adjust exploration and mining trends. An elite archiving mechanism is adopted to promote the convergence of the algorithm. Use crowding distance method to maintain the distribution of Pareto front. The balanced pool strategy is used to simulate the cooperative foraging behavior of slime mold, which helps to improve the exploration ability of the algorithm. Solve the latest CEC2020 functions.

       In order to verify the effectiveness of the proposed MOEOSMA, the CEC2020 function is used to analyze the convergence behavior of the algorithm in the target space and decision space. Unlike previous test suites, CEC2020 includes not only the true PF for each test problem, but also the associated local and global PS, allowing researchers to evaluate the performance of the algorithm in the target space and decision space. Since the CEC2020 function contains multiple globally optimal PSs, the good performance of the algorithm in the target space does not mean that multiple globally optimal PSs can be found. IGD (Zhang et al., 2008) in decision space (IGDX) and target space (IGDF) are used to evaluate the quality of the obtained PS and PF, respectively. In the decision space, the smaller the IGDX value, the closer the obtained PS is to the real PS. In the target space, the IGDF value is smaller. The closer the obtained PF is to the true PF.

1 Problem Definition and Evaluation Criteria for the CEC 2020 Special Conference on Multi-Mode Multi-Objective Optimization:

       In a multi-objective optimization problem, there may be two or more global or local Pareto optimal sets (PS), some of which may correspond to the same Pareto front (PF). These problems are defined as multimodal multiobjective optimization problems (MMOPs). Arguably, finding one of these multiple PSs may be enough to obtain an acceptable solution to some problems. However, failure to identify multiple PSs may prevent decision makers from considering solution options that could improve performance. Recently, many researchers have proposed different multi-mode multi-objective optimization (MMO) algorithms. In the MMO test suite of CEC'2020, a set of MMO experimental problems with different characteristics are designed, such as problems with PS and PF of different shapes, problems with local and global PS coexistence, scalable number of PS, decision variables and Target. In addition, a fair and appropriate evaluation standard and reference data are given to evaluate the performance of different MMO algorithms.

1.1 Related definitions:

       If a solution is not dominated by any other solution, it is called a non-dominated solution. The non-dominated solution set is called Pareto optimal set (PS). The set of vectors corresponding to PS in the target space is called Pareto front (PF).

       Global Pareto optimal set (Global PS): For any solution in the solution set GP, if there is no solution that dominates any solution in the set GP, then the GP is called a global Pareto optimal set.

       Global Pareto Front (Global PF): The set of all vectors in the target space corresponding to the global PS is defined as the global Pareto Front.

       Local Pareto optimal set (local PS): For any solution  in the solution set , if there is no neighborhood solution  that satisfies ( is a small positive value) and dominates any solution in the set, it is called a local Pareto optimal set;

        Local Pareto Front (local PF): The set of all vectors in the target space corresponding to the local PS is defined as the local Pareto Front.

       A method to determine whether a given multi-objective optimization problem is an MMO problem is given. For a multi-objective optimization problem, it is an MMO problem if it satisfies one of the following conditions:

1) It has at least one local Pareto optimal solution;

2) It has at least two global Pareto optimal solutions corresponding to the same point on PF.

       A solution that is not dominated by any neighborhood solution is called a local Pareto optimal solution. A solution that is not dominated by any solution in the feasible space is called a global Pareto optimal solution.

         Figure 1 shows a bi-objective minimization problem with two global PSs and one local PS. The solid line with stars represents global PS/PF, while the dashed line with dots represents local PS/PF. Note that a multi-modal multi-objective problem may have several local PSs and global PSs.

1.2 MOEOSMA solves CEC2020 multi-mode multi-objective function

MMF1：

MMF5: 

MMF7: 

 

function [fname,xl,xu,n_obj,n_var,repoint,N_ops] = func_info(Func_num)
% func_info: input the number of CEC2020 function and return the information about the corresponding function
switch Func_num
    case 1
        fname='MMF1';      % function name
        n_obj=2;           % the dimensions of the decision space
        n_var=2;           % the dimensions of the objective space
        xl=[1 -1];         % the low bounds of the decision variables
        xu=[3 1];          % the up bounds of the decision variables
        repoint=[1.1,1.1]; % reference point used to calculate the hypervolume, it is set to 1.1*(max value of f_i)
        N_ops=2;           % the number of global PS and local PS
    case 2
        fname='MMF2';
        n_obj=2;
        n_var=2;
        xl=[0 0];
        xu=[1 2];
        repoint=[1.1,1.1];
        N_ops=2;

.
.
.
.
 case 23
        fname='MMF16_l2';  % function name
        n_obj=3;
        n_var=3;
        xl=[0 0 0];
        xu=[1 1 1];
        repoint=[2.5,2.5,2.5];
        N_ops=3;
    case 24
        fname='MMF16_l3';  % function name
        n_obj=3;
        n_var=3;
        xl=[0 0 0];
        xu=[1 1 1];
        repoint=[2.5,2.5,2.5];
        N_ops=4;
end
end