Abstract

The classical Sard theorem (in a special case) states that the set of critical values of a C1C^1-map from an open set of Rn\mathbb{R}^n to Rn\mathbb{R}^n has Lebesgue measure zero. Motivated by a recent work of Barbet, Dambrine, Daniilidis and Rifford [Sard theorems for Lipschitz functions and applications in optimization, Israel J. Math. 212 (2016) 757-–790], we obtain in this paper versions of this theorem for a finite family of essentially smooth Lipschitz maps and for a locally Lipschitz continuous selection of this family. Here, a locally Lipschitz map is essentially smooth if its Clarke's subdifferential reduces to a singleton almost everywhere. As applications, we establish the genericity of Karush-Kuhn-Tucker type necessary condition for scalar/vector parametrized constrained optimization problems, Lebesgue zero measure of the set of Pareto optimal values of a map and the genericity of the finiteness of the solution set for a nonsmooth equation.

Author information

Contact details are reproduced from the original publication and may be historical.

Truong Xuan Duc Ha

Institute for Mathematics and Applied Sciences, Thang Long University, Hanoi, Vietnam

txdha@math.ac.vn

Keywords

  • Sard theorem
  • essentially smooth Lipschitz map
  • critical points
  • optimization
  • nonsmooth equation

Mathematics Subject Classification

49J40

References

  1. D. Azagra, J. Ferrera, J. Gomez-Gil: The Morse-Sard theorem revisited, Quart. J. Math. 69(3) (2018) 887–913.https://doi.org/10.1093/qmath/hay004Google Scholar
  2. L. Barbet, M. Dambrine, A. Daniilidis: The Morse-Sard theorem for Clarke critical values, Adv. Math. 242 (2013) 217–227.https://doi.org/10.1016/j.aim.2013.03.024Google Scholar
  3. L. Barbet, M. Dambrine, A. Daniilidis, L. Rifford: Sard theorems for Lipschitz functions and applications in optimization, Israel J. Math. 212(2) (2016) 757–790.https://doi.org/10.1007/s11856-016-1308-7Google Scholar
  4. S. M. Bates: Toward a precise smoothness hypothesis in Sard’s theorem, Proc. Amer. Math. Soc. 117(1) (1993) 279–283.https://doi.org/10.1090/S0002-9939-1993-1112486-4Google Scholar
  5. J.-M. Bonnisseau, J. Rivera-Cayupi: Constrained consumptions, Lipschitzian demands, and regular economies, J. Optimization Theory Appl. 131(2) (2006) 179–193.https://doi.org/10.1007/s10957-006-9147-zGoogle Scholar
  6. J. M. Borwein: Minimal cuscos and subgradients of Lipschitz functions, in: Fixed Point Theory and its Applications, J.-B. Baillion, M. Théra (eds.), Pitman Lecture Notes in Mathematics 252, Longman, Harlow (1991) 57–81.Google Scholar
  7. J. M. Borwein, W. B. Moors: Essentially smooth Lipschitz functions: compositions and chain rules, in: Foundations of Computational Mathematics, Rio de Janeiro 1997, Springer, Berlin (1997) 16–22.https://doi.org/10.1007/978-3-642-60539-0_2Google Scholar
  8. J. M. Borwein, W. B. Moors: Essentially smooth Lipschitz functions, J. Funct. Analysis 149(2) (1997) 305–351.https://doi.org/10.1006/jfan.1997.3101Google Scholar
  9. J. M. Borwein, W. B. Moors: Null sets and essentially smooth Lipschitz functions, SIAM J. Optimization 8(2) (1998) 309–323.https://doi.org/10.1137/S1052623496305213Google Scholar
  10. J. M. Borwein, X. Wang: Lipschitz functions with maximal subdifferentias are generic, Proc. Amer. Math. Soc. 128 (2000) 3221–3229.https://doi.org/10.1090/S0002-9939-00-05914-1Google Scholar
  11. F. Clarke: Generalized gradient and applications, Trans. Amer. Math. Soc. 205 (1975) 247–265.https://doi.org/10.1090/S0002-9947-1975-0367131-6Google Scholar
  12. F. Clarke: Optimization and Nonsmooth Analysis, Wiley, New York (1983).Google Scholar
  13. G. Debreu: Economies with a finite set of equilibria, Econometrica 38 (1970) 387–392.https://doi.org/10.2307/1909545Google Scholar
  14. H. Federer: Geometric Measure Theory, Springer, New York (1969).Google Scholar
  15. T. X. D. Ha: The Sard’s theorem for a class of locally Lipschitz mappings, Sém. d’Analyse Convexe 17, Exp. No. 9, Univ. du Languedoc, Montpellier (1989) 13 pp.Google Scholar
  16. R. Howard: Alexandrov’s theorem on the second derivatives of convex functions via Rademacher’s theorem on the first derivatives of Lipschitz functions, Notes from Lectures, available at http://people.math.sc.edu/howard/Notes/alex.pdf (1998).Google Scholar
  17. A. Jofre: Image of Critical Points Set for Nonsmooth Mappings and Applications, in: Contributions à l’Analyse Non-Différentiable avec Applications à l’Optimisation et à l’Analyse Non-Linéaire, Thèse, Université de Pau et des Pays de l’Adour, Pau, France (1989).Google Scholar
  18. E. Landis: On the functions representable as difference of two convex functions (Russian), Dokl. Akad. Nauk SSSR 80(1) (1951) 9–11.Google Scholar
  19. G. Lebourg: Generic differentiability of Lipschitz functions, Trans. Amer. Math. Soc. 256 (1979) 125–144.https://doi.org/10.1090/S0002-9947-1979-0546911-1Google Scholar
  20. A. P. Morse: The behavior of a function on its critical set, Ann. of Math. 40 (1939) 62–70.https://doi.org/10.2307/1968544Google Scholar
  21. I. P. Natanson: Theory of Functions of a Real Variable. Volume II, Frederick Ungar Publishing Co., New York (1957).Google Scholar
  22. A. Norton: The Zygmund Morse-Sard theorem, J. Geom. Analysis 4 (1994) 403–424.https://doi.org/10.1007/BF02921589Google Scholar
  23. B. H. Pourciau: Analysis and optimization of Lipschitz continuous mappings, J. Optimization Theory Appl. 22 (1977) 311–351.https://doi.org/10.1007/BF00932859Google Scholar
  24. L. Qi: Maximal normal operators and the second-order derivatives of nonsmooth functions, Industrial Engineering Technical Report 88-3, University of Wisconsin, Madison (1988).Google Scholar
  25. L. Qi: The maximal normal operator space and integration of subdifferentials of nonconvex functions, Nonlinear Analysis 13 (1989) 1003–1012.https://doi.org/10.1016/0362-546X(89)90091-6Google Scholar
  26. T. Rader: Nice demand functions, Econometrica 41 (1973) 913–935.https://doi.org/10.2307/1913814Google Scholar
  27. R. Saigal, C. Simon: Generic properties of the complementarity problem, Math. Programming 4 (1973) 324–335.https://doi.org/10.1007/BF01584674Google Scholar
  28. A. Sard: The measure of critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942) 883–890.https://doi.org/10.1090/S0002-9904-1942-07811-6Google Scholar
  29. A. Sard: Images of critical sets, Ann. of Math. 68 (1958) 247–259.https://doi.org/10.2307/1970246Google Scholar
  30. C. Shannon: Regular nonsmooth equations, J. Math. Economics 23 (1994) 147–165.https://doi.org/10.1016/0304-4068(94)90003-5Google Scholar
  31. J. Spingarn: Fixed and variable constraints in sensitivity analysis, SIAM J. Control Optimization 18(3) (1980) 297–310.https://doi.org/10.1137/0318021Google Scholar
  32. J. Spingarn: On optimality conditions for structured families of nonlinear programming problems, Math. Programming 22(1) (1982) 82–92.https://doi.org/10.1007/BF01581027Google Scholar
  33. J. Spingarn, R. T. Rockafellar: The generic nature of optimality conditions in nonlinear programming, Math. Oper. Res. 4(4) (1979) 425–430.https://doi.org/10.1287/moor.4.4.425Google Scholar

Suggested citation

T. X. D. Ha. “Versions of the Sard Theorem for Essentially Smooth Lipschitz Maps and Applications in Optimization and Nonsmooth Equations.” Journal of Convex Analysis 28 (2021), No. 1, 157–178. https://doi.org/10.68381/jca28012

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.