Parametric Borwein-Preiss variational principle and applications

Author:
Pando Gr. Georgiev

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3211-3225

MSC (2000):
Primary 49J35, 49J52; Secondary 46N10

DOI:
https://doi.org/10.1090/S0002-9939-05-07853-6

Published electronically:
June 20, 2005

MathSciNet review:
2161143

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Abstract: A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.

The tool for proving this parametric smooth variational principle is a useful lemma about continuous -minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

**1.**V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin,*Optimal Control*, Contemporary Soviet Mathematics, R. Garmkrelidze (ed.) Consultants Bureau, New York and London (1987). MR**0924574 (89e:49002)****2.**J.-P. Aubin, I. Ekeland,*Applied Nonlinear Analysis*, A Wiley - Interscience Publ., Jonh Wiley and Sons, 1984. MR**0749753 (87a:58002)****3.**J. Borwein and D. Preiss,*A smooth variational principle with applications to subdifferentiability and differentiability of convex functions*, Trans. Am. Math. Soc.,**303**(1987), 517-527. MR**0902782 (88k:49013)****4.**J. M. Borwein, Treiman, Jay S.; Zhu, Qiji J.*Partially smooth variational principles and applications*, Nonlinear Anal. 35 (1999), no. 8, Ser. B: Real World Applications, 1031-1059. MR**1707806 (2000j:49028)****5.**J. M. Borwein, Zhu, Qiji J.,*Variational analysis in nonreflexive spaces and applications to control problems with**perturbations*, Nonlinear Anal. 28 (1997), no. 5, 889-915 MR**1422192 (97k:49040)****6.**J. M. Borwein, Zhu, Qiji J.,*Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity*, SIAM J. Control Optim. 34 (1996), no. 5, 1568-1591. MR**1404847 (97g:49037)****7.**J. M. Borwein and A. S. Lewis,*Convex Analysis and Nonlinear Optimization. Theory and Examples*, CMS Books in Mathematics, Springer, 2000. MR**1757448 (2001h:49001)****8.**F. H. Clarke,*Optimization and Non-smooth Analysis*, J.Wiley and Sons, 1983.**9.**R. Deville and G. Godefroy and V. Zizler,*Un principle variationel utilisant des fonctions bosses*, C.R. Acad. Sci. Paris, Serie I,**312**(1991), 281-286. MR**1089715 (91j:49019)****10.**R. Deville and G. Godefroy and V. Zizler,*A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions*, J. Funct. Anal.**111**, (1993), 197-212. MR**1200641 (94b:49010)****11.**R. Deville and G. Godefroy and V. Zizler,*Smoothness and Renormings in Banach Spaces*, Pitman Monographs No.**64**, London: Longman, 1993. MR**1211634 (94d:46012)****12.**I. Ekeland,*On the variational principle*, J. Math. Anal. Appl.,**47**, (1974), 324-353. MR**0346619 (49:11344)****13.**I. Ekeland,*Nonconvex minimization problems*, Bull. Amer. Math. Soc.,**1**, (1979), 443-373. MR**0526967 (80h:49007)****14.**P. Gr. Georgiev,*The strong Ekeland variational principle, the strong drop theorem and applications*, J. Math. Anal. Appl.,**21**(1988), 1-21. MR**0934428 (89c:46019)****15.**S. Hu, N. Papageorgiou,*Handbook of Multivalued Analysis*, Vol. I, Kluwer, Mathematics and Its Applications, 419, 1997. MR**1485775 (98k:47001)****16.**E. Michael,*Continuous selections I*, Annals of Math.,**63**(1956), 361-382.MR**0077107 (17:990e)****17.**R. R. Phelps,*Convex functions, Monotone Operators and Differentiability*, Lecture Notes in Mathematics, No.**1364**.**18.**M. Sion,*On general minimax theorems*, Pacific J. Math.**8**(1958), 171-176. MR**0097026 (20:3506)**

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Additional Information

**Pando Gr. Georgiev**

Affiliation:
Department of Mathematics and Informatics, Sofia University “St. Kl. Ohridski", 5 James Bourchier Blvd., 1126 Sofia, Bulgaria

Address at time of publication:
Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, ML 0030, Cincinnati, Ohio 45221-0030

Email:
pgeorgie@ececs.uc.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07853-6

Keywords:
Borwein-Preiss variational principle,
Ky Fan's inequality,
continuous selections,
minimax problems.

Received by editor(s):
May 31, 1999

Published electronically:
June 20, 2005

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.