1. On the Computation of All Global Minimizers Through Particle Swarm Optimization IEEE Transactions On Evolutionary Computation, Vol. 8, No.3, June 2004 Konstantinos E. Parsopoulos and Michael N. Vrahatis Presented by Bo-Fu Liu 2005/09/07 Natural Computing Lab.,CSIE, NCTU

2. 1 Outline  Introduction  Particle Swarm Optimization Algorithm  Established Approach & Proposed Techniques  Experimental Results  Conclusion

3. 2 Introduction(1/2)  The minimization of multimodal functions with numerous local and global minima is a problem that frequently arises in diverse scientific fields.  There are some natural applications must to detect not just one just one, such as: computation of Nash equilibria in game theory (multi- objective optimization problem) computation of periodic orbits of nonlinear mappings

4. 3 Introduction(2/2)  Traditional adaptive stochastic search algorithms vs. Evolutionary Computation (EC) techniques.  The objective of this paper is detecting all global minimizers avoiding local minimizer

5. 4 Particle Swarm Optimization Algorithm For purposes of swarm intelligence: three heads are better than one.

6. 5 Particle Swarm Optimization Algorithm  Kennedy & Eberhart, 1995 Proposed the Particle Swarm Optimization Simulation of animal social behavior of finding food sources Not like the other population-based method directly transfer any information between individuals Easy implement & no need gradient informations

7. 6 Particle Swarm Optimization Algorithm Food Global Best Solution Past Best Solutio n

8. 7 Particle Swarm Optimization Algorithm － Movement Mechanism  Velocity calculation  Position update

9. 8 Particle Swarm Optimization Algorithm － Variant Particle Swarm Optimization  Clerc & Kennedy, 2002 Constriction factor

10. 9 Established Approach and the Proposed Techniques  Real-valued Optimization Problem unconstrained minimization task.  Local Optimization  Global Optimization f(x) = sin(x)^x/5e

11. 10 Established Approach and the Proposed Techniques  Traditional methods Simple: Multistart technique  Drawback: no mechanisms to prevent the previously detected minimizers Mathematical: Transformation technique  Drawback: not always feasible, computationally expensive Stochastic: Simulated annealing  One point vs. Population (ECs)

12. 11 Avoiding the local minimizer

13. 12 Approach 1 － Filled Function(1/5)  Transformation function  are arbitrary parameters  x * is a local minimum

14. 13 Approach 1 － Filled Function(2/5) x* = 4.60095589

15. 14 Approach 1 － Filled Function(3/5) x* = 4.60095589

16. 15 Approach 1 － Filled Function(4/5) x* = 4.60095589

17. 16 Approach 1 － Filled Function(5/5)  Drawback: Introduce new local minima at both side “mexican hat” Different problem needs different function Infeasible

18. 17 Approach 2 － Deflection Function(1/5)   The deflection technique is defined as exactly the same minimizers as f  Satisfy the aforementioned property

19. 18 Approach 2 － Deflection Function(2/5)  Example 1

20. 19 Approach 2 － Deflection Function(3/5)  Example 1

21. 20 Approach 2 － Deflection Function(4/5)  Example 2

22. 21 Approach 2 － Deflection Function(5/5)

23. 22 Approach 3 － Stretching Function(1/) 

24. 23 Approach 2 － Stretching Function(1/)

25. 24

26. 25 Repulsion Technique  Overcome the mexican hat problem

27. 26 Experimental Results － Numerical object function (1/3)  Computing Several Global Minimizers of Objective Functions Global minimizers:, Total have 12 global minimizers

28. 27 Experimental Results － Numerical object function (2/3)

29. 28 Experimental Results － Numerical object function (3/3)  Discussion Parameter  r ij : 0.1 (21/30), 4 (3/30)  p ij: 0.1 (12/22), 3 and 5 (22/22) Avoid local minimizer approach  The filled method failed Infeasible problem PSO parameter  Inertia factor vs. Constriction factor

30. 29 Alpha = 1, p=0.8 Experimental Results － Computing Periodic Orbits of Nonlinear Mappings (1/4)  Nonlinear mappings are used to model conservative or dissipative dynamical systems.  Central role in the analysis of such mappings is played by points, which are invariant under the mapping, called fixed points or periodic orbits. Alpha = 0.24, p=5 Alpha = 0.24

31. 30 Experimental Results － Computing Periodic Orbits of Nonlinear Mappings (2/4) Alpha = 0.24, p=5

32. 31 Experimental Results － Computing Periodic Orbits of Nonlinear Mappings (3/4) Alpha = 1, p=0.8

33. 32 Experimental Results － Computing Periodic Orbits of Nonlinear Mappings (4/4)  Traditional method can not overcome this kind of problem. Converge to fixed point Ignoring the very narrow channel

34. 33 Experimental Results － Computing Nash Equilibria in Finite Strategic Games (1/)  John Nash 1994, Nobelist Nash Equilibria are sets of strategies for players in a noncooperative game such that no single one of them would be better off switching strategies unless others did.

35. 34 Experimental Results － Computing Nash Equilibria in Finite Strategic Games (2/)

36. 35 Conclusion  This paper proposes approaches for effectively computing all global minimiers.  Incorporating deflection and stretching techniques overcome local minimizers, as well as a repulsion technique to repel particles away from previously detected minimizers.  Experimental results indicate that this is an effective approach for computing more than one global minimizers that can be used to solve high-edge problems of nonlinear science and economic theory.