Abstract

We derive criteria for upper Lipschitz/Hölder continuity of the set of minimal points of a given subset A⊂YA\subset Y of a normed space Y when A is subjected to perturbations. To this aim we introduce the rate of containment of A, a real-valued function of one real variable, which measures the depart from minimality as a function of the distance from the minimal point set. The main requirement we impose is that for small arguments the rate of containment is a sufficiently fast growing function. The obtained results are applied to parametric vector optimization problems to derive conditions for upper Hölder continuity of the performance multifunction.

Author information

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

Ewa M. Bednarczuk

Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warszawa, Poland

bednarcz@ibspan.waw.pl

Keywords

  • Minimal points
  • Hölder multivalued mappings
  • parametric vector optimization

References

  1. H. Attouch, R. Wets: Quantitative Stability of Variational systems: I. The epigraphical distance, Transactions of the AMS 328 (1991) 695–729.https://doi.org/10.1090/S0002-9947-1991-1018570-0Google Scholar
  2. H. Attouch, R. Wets: Quantitative Stability of Variational systems: II. A framework for nonlinear conditioning, IIASA Working paper 88-9, Laxemburg, Austria, February 1988.Google Scholar
  3. H. Attouch, R. Wets: Lipschitzian stability of the ε−approximate solutions in convex optimization, SIAM Journal on Optimization 3 (1992) 359–381.https://doi.org/10.1137/0803016Google Scholar
  4. J.-P. Aubin, H. Frankowska: Set-valued Analysis, Birkhäuser (1990).Google Scholar
  5. E. M. Bednarczuk: Berge-type theorems for vector optimization problems, Optimization 32 (1995) 373–384.https://doi.org/10.1080/02331939508844057Google Scholar
  6. E. M. Bednarczuk: Stability results for vector optimization problems in partially ordered topological vector spaces, in: Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, August 1992, V. Lakshmikantham (ed.), Walter de Gruyter, Berlin, New York (1996) 2371–2382.https://doi.org/10.1515/9783110883237.2371Google Scholar
  7. E. M. Bednarczuk: An approach to well-posedness in vector optimization: consequences to stability, Control and Cybernetics 23 (1994) 107–122.Google Scholar
  8. E. M. Bednarczuk, W. Song: PC points and their application to vector optimization, Pliska Studia Mathematica Bulgarica 12 (1998) 21–30.Google Scholar
  9. N. Bolintineanu, A. El-Maghri: On the sensitivity of efficient points, Revue Roumaine de Mathematiques Pures et Appliques 42 (1997) 375–382.Google Scholar
  10. M. P. Davidson: Lipschitz continuity of Pareto optimal extreme points, Vestnik Mosk. Univ., Ser. XV, Vychisl. Mat. Kiber. 63 (1996) 41–45.Google Scholar
  11. M. P. Davidson: Conditions for stability of a set of extreme points of a polyhedron and their applications, Russ. Akad. Nauk, Vychisl. Tsentr, Moscow (1996).https://doi.org/10.1007/BF02190003Google Scholar
  12. M. P. Davidson: On the Lipschitz stability of weakly Slater systems of convex inequalities, Vestnik Mosk. Univ., Ser. XV (1998) 24–28.Google Scholar
  13. Deng-Sien: On approximate solutionss in convex vector optimization, SIAM Journal on Control and Optimization 35 (1997) 2128–2136https://doi.org/10.1137/S0363012995292561Google Scholar
  14. A. Dontchev, T. Rockafellar: Characterization of Lipschitzian stability, in: Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker (1998) 65–82.https://doi.org/10.1201/9781003072119-4Google Scholar
  15. J. Jahn: Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt am Main (1986).Google Scholar
  16. R. Janin, J. Gauvin: Lipschitz dependence of the optimal solutions to elementary convex programs, in: Proceedings of the 2nd Catalan Days on Applied Mathematics, Presses University, Perpignan (1995).Google Scholar
  17. Li-Wu: Error bounds for piecewise convex quadratic programs and applications, SIAM Journal on Control and Optimization 33 (1995) 1510–1529.https://doi.org/10.1137/S0363012993243022Google Scholar
  18. D. T. Luc: Theory of Vector Optimization, Springer-Verlag, Berlin, Heidelberg, New York (1989).https://doi.org/10.1007/978-3-642-50280-4Google Scholar
  19. B. Mordukhovich: Sensitivity analysis for constraints and variational systems by means of set-valued differentiation, Optimization 31 (1994) 13–43.https://doi.org/10.1080/02331939408844004Google Scholar
  20. B. Mordukhovich, Yong Heng Shao: Differential characterisations of convering, metric regularity and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Analysis, Theory, Methods and Applications 25 (1995) 1401–1424.https://doi.org/10.1016/0362-546X(94)00256-HGoogle Scholar
  21. S. M. Robinson: Generalized equations and their solutions, part I: Basic theory, Mathematical Programming Study 10 (1979) 128–141.https://doi.org/10.1007/BFb0120850Google Scholar
  22. S. M. Robinson: A characterization of stability in linear programming, Operations Research 25 (1977) 435–447.https://doi.org/10.1287/opre.25.3.435Google Scholar
  23. S. M. Robinson: Stability of systems of inequalities, part II: Differentiable nonlinear systems, SIAM J. Numerical Analysis 13 (1976) 497–513.https://doi.org/10.1137/0713043Google Scholar
  24. R. T. Rockafellar: Lipschitzian properties of multifunctions, Nonlinear Analysis, Theory, Methods and Applications 9 (1985) 867–885.https://doi.org/10.1016/0362-546X(85)90024-0Google Scholar
  25. R. T. Rockafellar, J.-B. Wets: Variational systems, an introduction, in: Multifunctions and Integrands, Stochastic Analysis, Approximation and Optimization, Proceedings, Catania 1983, G. Salinetti (ed.), Springer-Verlag, Berlin (1984).Google Scholar
  26. S. Rolewicz, D. Pallaschke: Foundations of Mathematical Optimization, Kluwer (1998).https://doi.org/10.1007/978-94-017-1588-1Google Scholar
  27. N. D. Yen: Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint, Mathematics of Oper. Res. 20 (1995) 695–705.https://doi.org/10.1287/moor.20.3.695Google Scholar
  28. X. Q. Yang: Directional derivatives for set-valued mappings and applications, Mathematical Methods of Oper. Res. 48 (1998) 273–283.https://doi.org/10.1007/s001860050028Google Scholar

Suggested citation

E. M. Bednarczuk. “Upper Hölder Continuity of Minimal Points.” Journal of Convex Analysis 9 (2002), No. 2, 327–338. https://doi.org/10.68381/jca09020

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.