1. Locating Multiple Optimal Solutions Based on Multiobjective Optimization Yong Wang ywang@csu.edu.cn

2. 2  Part I: Application to Nonlinear Equation Systems (MONES)  Part II: Application to Multimodal Optimization Problems (MOMMOP)  Future Work Outline of My Talk

3. 3  Part I: Application to Nonlinear Equation Systems (MONES)  Part II: Application to Multimodal Optimization Problems (MOMMOP)  Future Work Outline of My Talk

4. 4 Nonlinear Equation Systems (NESs) (1/2) NESs arise in many science and engineering areas such as chemical processes, robotics, electronic circuits, engineered materials, and physics. The formulation of a NES

5. Nonlinear Equation Systems (NESs) (2/2) An example 5 the optimal solutions A NES may contain multiple optimal solutions

6. 6 Solving NESs by Evolutionary Algorithms (1/4) The aim of solving NESs by evolutionary algorithms (EAs) –Locate all the optimal solutions in a single run At present, there are three kinds of methods –Single-objective optimization based methods –Constrained optimization based methods –Multiobjective optimization based methods

7. 7 Solving NESs by Evolutionary Algorithms (2/4) Single-objective optimization based methods The main drawback –Usually, only one optimal solution can be found in a single run or

8. Solving NESs by Evolutionary Algorithms (3/4) Constrained optimization based methods The main drawbacks –Similar to the first kind of method, this kind of methods can only locate one optimal solution in a single run –Additional constraint-handling techniques should be integrated 8 or

9. Solving NESs by Evolutionary Algorithms (4/4) Multiobjective optimization based methods (CA method) The main drawbacks –It may suffer from the “curse of dimensionality” (i.e., many- objective) –Maybe only one solution can be found in a single run 9 C. Grosan and A. Abraham, “A new approach for solving nonlinear equation systems,” IEEE Transactions on Systems Man and Cybernetics - Part A, vol. 38, no. 3, pp. 698- 714, 2008.

10. 10 MONES: Multiobjective Optimization for NESs (1/8) The main motivation –When solving a NES by EAs, it is expected to locate multiple optimal solutions in a single run. –Obviously, the above process is similar to that of the solution of multiobjective optimization problems by EAs. –A question arises naturally is whether a NES can be transformed into a multiobjective optimization problems and, as a result, multiobjective EAs can be used to solve the transformed problem.

11. MONES: Multiobjective Optimization for NESs (2/8) Multiobjective optimization problems –Pareto dominance –Pareto optimal solutions The set of all the nondominated solutions –Pareto front The images of the Pareto optimal solutions in the objective space ≤ < Pareto dominates minimize ≤≤ ≤ f1f1 f2f2 11

12. 12 MONES: Multiobjective Optimization for NESs (3/8) The main idea ① ② minimize

13. 13 MONES: Multiobjective Optimization for NESs (4/8) The principle of the first term The images of the optimal solutions of the first term in the objective space are located on the line of ‘y=1-x’ minimize

14. 14 MONES: Multiobjective Optimization for NESs (5/8) The principle of the second term minimize

15. 15 MONES: Multiobjective Optimization for NESs (6/8) The principle of the first term plus the second term The images of the optimal solutions of a NES in the objective space are located on the line of ‘y=1-x’ minimize Pareto Front 0 1 1

16. 16 MONES: Multiobjective Optimization for NESs (7/8) The differences between MONES and CA CAMONES

17. 17 MONES: Multiobjective Optimization for NESs (8/8) The differences between MONES and CA CA MONES

18. 18  Part I: Application to Nonlinear Equation Systems (MONES)  Part II: Application to Multimodal Optimization Problems (MOMMOP)  Future Work Outline of My Talk

19. Multimodal Optimization Problems (MMOPs) (1/2) Many optimization problems in the real-world applications exhibit multimodal property, i.e., multiple optimal solutions may coexist. The formulation of multimodal optimization problems (MMOPs) 19

20. Multimodal Optimization Problems (MMOPs) (2/2) Several examples 20

21. 21 The Previous Work (1/2) Niching methods –The first niching method The preselection method suggested by Cavicchio in 1970 –The current popular niching methods Clearing (Pétrowski, ICEC, 1996) Sharing (Goldberg and Richardson, ICGA, 1987) Crowding (De Jong, PhD dissertation, 1975) restricted tournament selection (Harik, ICGA, 1995) Speciation (Li et al., ECJ, 2002) The main disadvantages –Some problem-dependent niching parameters are required

22. The Previous Work (2/2) Multiobjective optimization based methods, usually two objectives are considered: –The first objective: the original multimodal function –The second objective: the distance information (Das et al., IEEE TEVC, 2013) or the gradient information (Deb and Saha, ECJ, 2012) The disadvantages –It cannot guarantee that the two objectives in the transformed problem totally conflict with each other –The relationship between the optimal solutions of the original problems and the Pareto optimal solutions of the transformed problems is difficult to be verified theoretically. 22

23. 23 MOMMOP: Multiobjective Optimization for MMOPs (1/5) The main motivation minimize

24. 24 MOMMOP: Multiobjective Optimization for MMOPs (2/5) The main idea –Convert an MMOP into a biobjective optimization problem ① ② minimize

25. 25 MOMMOP: Multiobjective Optimization for MMOPs (3/5) The principle of the second term the objective function value of the best individual found during the evolution the objective function value of the worst individual found during the evolution the objective function value of the current individual the range of the first variable Remark: the aim is to make the first term and the second term have the same scale the scaling factor For the optimal solutions of the original multimodal optimization problems, the values of the second term are equal to zero.

26. 26 MOMMOP: Multiobjective Optimization for MMOPs (4/5) The principle of the first term plus the second term The images of the optimal solutions of a multimodal optimization problem in the objective space are located on the line of ‘y=1-x’ minimize

27. 27 MOMMOP: Multiobjective Optimization for MMOPs (5/5) Why does MOMMOP work? –MOMMOP is an implicit niching method x f(x)f(x) xbxb xaxa xcxc (0.1, 1) (0.15, 0.8) (0.6, 0.8) f1f1 f2f2 x f(x)f(x) 0 1 1 1 1 0 0 1 1 1 1 1 1 (0.0, 0.9) (0.35, 1.05) (0.8, 0.6)

28. 28 Two issues in MOMMOP (1/2) The first issue –Some optimal solutions may have the same value in one or many decision variables

29. 29 Two issues in MOMMOP (2/2) The second issue –In some basins of attraction, maybe there are few individuals

30. 30  Part I: Application to Nonlinear Equation Systems (MONES)  Part II: Application to Multimodal Optimization Problems (MOMMOP)  Future Work Outline of My Talk

31. 31 Future Work We proposed two similar frameworks for nonlinear equation systems and multimodal optimization problems, respectively, however –The principle should be analyzed in depth in the future –The rationality should be further verified –The frameworks could be improved Our frameworks could be generalized into solve other kinds of optimization problems