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

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Presentation on theme: "4 Mei 2009Universitaet WIEN1 Stability of Equilibrium Points Roger J-B Wets, Univ. California, Davis with Alejandro Jofr é, Universidad de Chile."— Presentation transcript:

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) = L = -1 L =0 L = -1 hypo/epilopsided

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

45 4 Mei 2009Universitaet WIEN45


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