1. 4 Mei 2009Universitaet WIEN1 Stability of Equilibrium Points Roger J-B Wets, Univ. California, Davis with Alejandro Jofr é, Universidad de Chile

2. 4 Mei 2009Universitaet WIEN2 Generalized Equations: Approximate Solutions

3. 4 Mei 2009Universitaet WIEN3 Variational Inequalities non-empty, convex continuous find where approximating solutions? with

4. 4 Mei 2009Universitaet WIEN4 In principle: approach

5. 4 Mei 2009Universitaet WIEN5 Aubin ’ s ineluctable circle

6. 4 Mei 2009Universitaet WIEN6 V.I.: Our approach Let then if and only if

7. 4 Mei 2009Universitaet WIEN7 V.I.:The approach

8. 4 Mei 2009Universitaet WIEN8 Fixed Points (Brouwer) Our approach

9. 4 Mei 2009Universitaet WIEN9 Non-Cooperative Games Nash equilibrium: such that is a Nash equilibrium

10. 4 Mei 2009Universitaet WIEN10 The approach:

11. 4 Mei 2009Universitaet WIEN11 Applications: Convergence and stability Saddle-points: Lagrangians, Hamiltonians Fixed points Solutions of cooperative and non-cooperative games Economic-Equilibrium points (Walras) Generalized Nash Equilibrium Problems Set-valued inclusions, generalized equations Stability of mountain-pass paths & minimal surfaces (bi-topologies)

12. 4 Mei 2009Universitaet WIEN12 Optimization: Max-Framework C f0f0

13. 4 Mei 2009Universitaet WIEN13 Hypo-Convergence (max-framework) argmax f a  argmax f argmax (f + g)  argmax (f+g),  cont. g hypo-convergence

14. 4 Mei 2009Universitaet WIEN14 Hypo-Convergence argmax g a ~ argmax g

15. 4 Mei 2009Universitaet WIEN15 Hypo-convergence: Properties (  -convergence)

16. 4 Mei 2009Universitaet WIEN16 Tight Hypo-Convergence

17. 4 Mei 2009Universitaet WIEN17 Lopsided convergence max-inf framework

18. 4 Mei 2009Universitaet WIEN18 Lopsided convergence: definition

19. 4 Mei 2009Universitaet WIEN19 Lopsided: elementary properties a argmax-inf K a ~ argmax-inf K lopsided epi-convergence if lopsided hypo-convergence if Remarks: not a  -convergence, in  -dim. different topologies

20. 4 Mei 2009Universitaet WIEN20 Lopsided ancillary-tightly

21. 4 Mei 2009Universitaet WIEN21 Proof …. then apply

22. 4 Mei 2009Universitaet WIEN22 Ky Fan functions & inequality

23. 4 Mei 2009Universitaet WIEN23 Extending Ky Fan ’ s inequality gener.-Ky Fan fcns closed under lopsided saddle fcns closed under h/e-convergence usc fcns closed under hypo-convergence

24. 4 Mei 2009Universitaet WIEN24 APPLICATIONS

25. 4 Mei 2009Universitaet WIEN25 Fixed Points (Brouwer) Ky Fan Inequality applies (usc,convex, K(x,x) ≥ 0 ):

26. 4 Mei 2009Universitaet WIEN26 Convergence of fixed points Hence Extension: C just convex

27. 4 Mei 2009Universitaet WIEN27 Linear Complementarity Problems

28. 4 Mei 2009Universitaet WIEN28 Set-valued Inclusions (sufficient)

29. 4 Mei 2009Universitaet WIEN29 Variational Inequalities C   n nonempty, convex, compact G : C   m continuous, (m=n) find where with K is a Ky Fan function, K(u,u) ≥ 0. Find

30. 4 Mei 2009Universitaet WIEN30 Convergence: V.I. i.e., Ky Fan functions lop- converges ancillary-tight to K, i.e. any cluster point of the solutions of the approximating V.I. is sol ’ n of limit V.I. (sufficient conditions) + extension

31. 4 Mei 2009Universitaet WIEN31 Nash Equilibrium points

32. 4 Mei 2009Universitaet WIEN32 Walras Equilibrium points Walrasian: Ky Fan fcn conditions: Convergence:

33. 4 Mei 2009Universitaet WIEN33 Numerical Schemes Walras Equilibrium points

34. 4 Mei 2009Universitaet WIEN34 Augmented Walrasian A. Bagh & S. Lucero (2002, UC-Davis)-deterministic J. Deride, A. Jofr é & R.W. (2008-9, CMM)-stochastic

35. 4 Mei 2009Universitaet WIEN35 Iterations experiments: 10 agents, 150 goods (easy!) minimizing a linear form on a ball reduces to finding the largest element of s(p k )

36. 4 Mei 2009Universitaet WIEN36 Test: Demand functions Cobb-Douglas utility function: budget constraint: demand: (demand = supply)

37. 4 Mei 2009Universitaet WIEN37 SADDLE-POINTS

38. 4 Mei 2009Universitaet WIEN38 Lagrangians (& Hamiltonians) Lagrangian: concave-convex, max-inf framework

39. 4 Mei 2009Universitaet WIEN39 Hypo/Epi-Convergence a saddle point L a ~ saddle point L (90 ’ s) definition: “ problem ” : hypo/epi-topology not Hausdorff …. the pitfall “ equivalence classes ” dom K    

40. 4 Mei 2009Universitaet WIEN40 Hypo/Epi convergence like lopsided convergence not like lopsided convergence

41. 4 Mei 2009Universitaet WIEN41 Hypo/epi: elementary properties a saddle points L a ~ saddle points L hypo/epi epi-convergence if hypo/epi hypo-convergence if Remarks: not a  -convergence, in  -dim. different topologies but not y -epi-convergence and x -hypo-convergence

42. 4 Mei 2009Universitaet WIEN42 Hypo/epi-convergence: Properties & Applications Applications: Convergence of solutions to zero-sum games Joint convergence of solutions to dual convex programs Mechanics

43. 4 Mei 2009Universitaet WIEN43 Lopsided & hypo/epi-convergence lopsided hypo/epi-convergence not conversely (-1,-1) L(0,0) = 0 - -1 L = -1 L =0 L = -1 hypo/epilopsided

44. 4 Mei 2009Universitaet WIEN44 Lopsided & hypo/epi-convergence

45. 4 Mei 2009Universitaet WIEN45