A prevalent transversality theorem for Lipschitz functions
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Abstract:
This paper provides a version of the transversality theorem for a class of Lipschitz functions of the form $f:\textbf {R}^n\times C\to \textbf {R}^n$, where $C$ is a convex subset of a normed vector space $Z$ indexing the parameters in the problem. The set $C$ may be infinite-dimensional, and the notion of generic used is the measure-theoretic notion of prevalence introduced by Hunt, Sauer and Yorke (1992) and Christensen (1974). This paper also provides some results on sensitivity analysis for solutions to locally Lipschitz equations.References
- Robert M. Anderson and William R. Zame, Genericity with infinitely many parameters, Adv. Theor. Econ. 1 (2001), Art. 1, 64. MR 2002579, DOI 10.2202/1534-5963.1003
- Jean-Pierre Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), no. 1, 87–111. MR 736641, DOI 10.1287/moor.9.1.87
- J. P. R. Christensen, Topology and Borel structure, North-Holland Mathematics Studies, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory. MR 0348724
- Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
- F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998. MR 1488695
- Gerard Debreu, Economies with a finite set of equilibria, Econometrica 38 (1970), 387–392. MR 278702, DOI 10.2307/1909545
- Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810
- M. K. Fort Jr., Essential and non essential fixed points, Amer. J. Math. 72 (1950), 315–322. MR 34573, DOI 10.2307/2372035
- Brian R. Hunt, Tim Sauer, and James A. Yorke, Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. MR 1161274, DOI 10.1090/S0273-0979-1992-00328-2
- Alan J. King and R. Tyrrell Rockafellar, Sensitivity analysis for nonsmooth generalized equations, Math. Programming 55 (1992), no. 2, Ser. A, 193–212. MR 1167597, DOI 10.1007/BF01581199
- Trout Rader, Nice demand functions, Econometrica 41 (1973), 913–935. MR 441276, DOI 10.2307/1913814
- R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 449–482. Analyse non linéaire (Perpignan, 1987). MR 1019126, DOI 10.1016/S0294-1449(17)30034-3
- R. T. Rockafellar, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806, DOI 10.1090/S0002-9947-1988-0936806-9
- R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
- Chris Shannon, Regular nonsmooth equations, J. Math. Econom. 23 (1994), no. 2, 147–165. MR 1266510, DOI 10.1016/0304-4068(94)90003-5
- Chris Shannon, Determinacy of competitive equilibria in economies with many commodities, Econom. Theory 14 (1999), no. 1, 29–87. MR 1707170, DOI 10.1007/s001990050282
- Shannon, C. (2002), “Inada Conditions and the Determinacy of Equilibria,” preprint, October 2002.
- Chris Shannon and William R. Zame, Quadratic concavity and determinacy of equilibrium, Econometrica 70 (2002), no. 2, 631–662. MR 1913825, DOI 10.1111/1468-0262.00298
- Maxwell B. Stinchcombe, The gap between probability and prevalence: loneliness in vector spaces, Proc. Amer. Math. Soc. 129 (2001), no. 2, 451–457. MR 1694881, DOI 10.1090/S0002-9939-00-05543-X
Additional Information
- Chris Shannon
- Affiliation: Department of Economics and Department of Mathematics, University of California–Berkeley, Berkeley, California 94720
- Email: cshannon@econ.berkeley.edu
- Received by editor(s): April 25, 2003
- Published electronically: April 13, 2006
- Additional Notes: Thanks to Bob Anderson, Don Brown, Max Stinchcombe and Bill Zame for helpful comments and conversations concerning this paper. The financial support of the National Science Foundation under grant SBR 98-18759, the Miller Institute, and an Alfred P. Sloan Foundation Research Fellowship is gratefully acknowledged.
- Communicated by: Jonathan M. Borwein
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2755-2765
- MSC (2000): Primary 58C05, 49J52, 90C31; Secondary 58E17
- DOI: https://doi.org/10.1090/S0002-9939-06-08607-2
- MathSciNet review: 2213756