Parametric Borwein-Preiss variational principle and applications

Georgiev, Pando

doi:10.1090/S0002-9939-05-07853-6

Parametric Borwein-Preiss variational principle and applications
HTML articles powered by AMS MathViewer

by Pando Gr. Georgiev PDF

Proc. Amer. Math. Soc. 133 (2005), 3211-3225 Request permission

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 $\varepsilon$-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.

References

V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. Translated from the Russian by V. M. Volosov. MR 924574, DOI 10.1007/978-1-4615-7551-1
Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), no. 2, 517–527. MR 902782, DOI 10.1090/S0002-9947-1987-0902782-7
Jonathan M. Borwein, Jay S. Treiman, and Qiji J. Zhu, Partially smooth variational principles and applications, Nonlinear Anal. 35 (1999), no. 8, Ser. B: Real World Applications, 1031–1059. MR 1707806, DOI 10.1016/S0362-546X(98)00113-8
Jonathan M. Borwein and Qiji J. Zhu, Variational analysis in nonreflexive spaces and applications to control problems with $L^1$ perturbations, Nonlinear Anal. 28 (1997), no. 5, 889–915. MR 1422192, DOI 10.1016/0362-546X(95)00186-Y
Jonathan M. Borwein and Qiji J. Zhu, 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, DOI 10.1137/S0363012994268801
Jonathan M. Borwein and Adrian S. Lewis, Convex analysis and nonlinear optimization, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 3, Springer-Verlag, New York, 2000. Theory and examples. MR 1757448, DOI 10.1007/978-1-4757-9859-3
F. H. Clarke, Optimization and Non-smooth Analysis, J.Wiley and Sons, 1983.
Robert Deville, Gilles Godefroy, and Václav Zizler, Un principe variationnel utilisant des fonctions bosses, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 3, 281–286 (French, with English summary). MR 1089715
Robert Deville, Gilles Godefroy, and Václav Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), no. 1, 197–212. MR 1200641, DOI 10.1006/jfan.1993.1009
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443–474. MR 526967, DOI 10.1090/S0273-0979-1979-14595-6
Pando Grigorov Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988), no. 1, 1–21. MR 934428, DOI 10.1016/0022-247X(88)90187-4
Shouchuan Hu and Nikolas S. Papageorgiou, Handbook of multivalued analysis. Vol. I, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, 1997. Theory. MR 1485775, DOI 10.1007/978-1-4615-6359-4
Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
R. R. Phelps, Convex functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, No. 1364.
Maurice Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171–176. MR 97026, DOI 10.2140/pjm.1958.8.171