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Strong stability in nonlinear programming revisited

Published online by Cambridge University Press:  17 February 2009

Diethard Klatte  and
Bernd Kummer
Diethard Klatte
Affiliation:
Institut für Operations Research, Universität Zürich, Moussonstr. 15, CH-8044 Zurich, Switzerland
Bernd Kummer
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
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Abstract

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The paper revisits characterizations of strong stability and strong regularity of KarushKuhn-Tucker solutions of nonlinear programs with twice differentiable data. We give a unified framework to handle both concepts simultaneously.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 3 , January 1999 , pp. 336 - 352
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bank, B., Guddat, J., Klatte, D.Kummer, B. and Tammer, K., Non-Linear Parametric Optimization (Akademie, Berlin, 1982).CrossRefGoogle Scholar
[2]Bonnans, F. and Shapiro, A., “Optimization problems with perturbations, a guided tour”, SIAM Review 40 (1998) 228–264.CrossRefGoogle Scholar
[3]Bonnans, F. and Sulem, A., “Pseudopower expansion of solutions of generalized equations and constrained optimization problems”, Math. Programming 70 (1995) 123–148.CrossRefGoogle Scholar
[4]Clarke, F., “On the inverse function theorem”, Pacific J. Math. 64 (1976) 97–102.CrossRefGoogle Scholar
[5]Clarke, F., Optimization and Nonsmooth Analysis (Wiley, New York, 1983).Google Scholar
[6]Dontchev, A., “A proof of the necessity of LI and SSOSC for Lipschitzian stability in nonlinear programming”, preprint Math. Reviews, Ann Arbor, M148104, USA, May 1997.Google Scholar
[7]Dontchev, A. and Rockafellar, R. T., “Characterizations of strong regularity for variational inequalities over polyhedral convex sets”, SIAM J. Optimization 6 (1996) 1087–1105.CrossRefGoogle Scholar
[8]Dontchev, A. and Rockafellar, R. T., “Characterizations of Lipschitz stability in nonlinear programming”, in: Mathematical Programming with Data Perturbations (ed. Fiacco, A. V.), (Marcel Dekker, New York, 1997) 65–82.Google Scholar
[9]Gfrerer, H., “Hölder continuity of solutions of perturbed optimization problems under MangasarianFromovitz Constraint Qualification”, in: Parametric Optimization and Related Topics (eds. Kummer, B., Guddat, J., Jongen, H. Th. and Nožička, F.), (Akademie, Berlin, 1987) 113–124.Google Scholar
[10]Jongen, H. Th., Klatte, D. and Tammer, K., “Implicit functions and sensitivity of stationary points”, Math. Programming 49 (1990) 123–138.CrossRefGoogle Scholar
[11]Jongen, H. Th., Möbert, T., Ruckmann, J. and Tammer, K., “On inertia and Schur complement in optimization”. Linear Algebra Appl. 95 (1987) 97–109.CrossRefGoogle Scholar
[12]Jongen, H. Th., Möbert, T. and Tammer, K., “On iterated minimization in nonconvex optimization”, Math. Open Res. 11 (1986) 679–691.CrossRefGoogle Scholar
[13]King, A. and Rockafellar, R. T., “Sensitivity analysis for nonsmooth generalized equations”, Math. Programming 55 (1992) 341–364.CrossRefGoogle Scholar
[14]Klatte, D. and Kummer, B., “Stability properties of infima and optimal solutions of parametric optimization problems”, in: Nondifferentiable Optimization: Motivations and Applications (eds. Demyanov, V. F. and Pallaschke, D.), (Springer, Berlin, 1985) 215–229.CrossRefGoogle Scholar
[15]Klatte, D. and Kummer, B., “Generalized Kojima-functions and Lipschitz stability of critical points”, manuscript, Institut Operations Research, Universität Zürich, and Institut für Mathematik, Humboldt-Universität, Berlin, December 1997.Google Scholar
[16]Klatte, D. and Tammer, K., “Strong stability of stationary solutions and Karush-Kuhn-Tucker points in nonlinear optimization”, Ann. Oper. Res. 27 (1990) 285–308.CrossRefGoogle Scholar
[17]Kojima, M., “Strongly stable stationary solutions on nonlinear programs”, in: Analysis and Computation of Fixed Points (ed. Robinson, S. M.), (Academic Press, New York, 1980) 93–138.CrossRefGoogle Scholar
[18]Kummer, B., “ The inverse of a Lipschitz function in Rn: Complete characterization by directional derivatives”, Preprint No. 195, (Humboldt-Universität, Sektion Mathematik, Berlin, 1988).Google Scholar
[19]Kummer, B., “Lipschitzian inverse functions, directional derivatives and application in C1.1 optimization”, J. Optim. Theory Appl 70 (1991) 559–580.CrossRefGoogle Scholar
[20]Kummer, B., “An implicit function theorem for C0.1-equations and parametric C1.1-optimization”, J. Math. Analysis Appl. 158 (1991) 35–46.CrossRefGoogle Scholar
[21]Kummer, B., “Lipschitzian and pseudo-Lipschitzian inverse functions and applications to nonlinear programming”, in: Mathematical Programming with Data Perturbations (ed. Fiacco, A. V.), (Marcel Dekker, New York, 1997) 201–222.Google Scholar
[22]Levy, A. B., “Implicit multifunction theorems for the sensitivity analysis of variational conditions”, Math. Programming 74 (1996) 333–350.CrossRefGoogle Scholar
[23]Levy, A. B. and Rockafellar, R. T., “Sensitivity of solutions in nonlinear programs with nonunique multiplier”, in: Recent Advances in Nonsmooth Optimzation (eds. Du, D. Z., Qi, L. and Womersley, R. S.), (World Scientific Press, 1995) 215–223.CrossRefGoogle Scholar
[24]Liu, J., “Strong stability in variational inequalities”, SIAM J. Control Optim. 33 (1995) 725–749.CrossRefGoogle Scholar
[25]Pang, J. -S., “Necessary and sufficient conditions for solution stability of parametric nonsmooth equations”, in: Recent Advances in Nonsmooth Optimization (eds. Du, D. Z., Qi, L. and Womersley, R. S.), (World Scientific Press, 1995) 261–288.CrossRefGoogle Scholar
[26]Pang, J. -S. and Ralph, D., “Piecewise smoothness, local invertibility, and parametric analysis of normal maps”, Math. Oper. Res. 21 (1996) 401–426.CrossRefGoogle Scholar
[27]Ralph, D. and Dempe, S., “Directional derivatives of the solution of a parametric nonlinear program”, Math. Programming 70 (1995) 159–172.CrossRefGoogle Scholar
[28]Ralph, D. and Scholtes, S., “Sensitivity analysis of composite piecewise smooth equations”, Math. Programming B 76 (1997) 593–612.CrossRefGoogle Scholar
[29]Robinson, S. M., “Strongly regular generalized equations”, Math. Oper. Res. 5 (1980) 43–62.CrossRefGoogle Scholar
[30]Robinson, S. M., “Generalized equations and their solutions, Part II: Applications to nonlinear programming”, Math. Programming Study 19 (1982) 200–221.CrossRefGoogle Scholar
[31]Robinson, S. M., “An implicit function theorem for a class of nonsmooth functions”, Math. Oper. Res. 16 (1991) 292–309.CrossRefGoogle Scholar
[32]Robinson, S. M., “Normal maps induced by linear transformations”, Math. Oper. Res. 17 (1992) 691–714.CrossRefGoogle Scholar
[33]Thibault, L., “On generalized differentials and subdifferentials of Lipschitz vector-valued functions”, Nonlinear Anal.: Theory, Methods, and Appl. 6 (1982) 1037–1053.CrossRefGoogle Scholar