Abstract
In a recent paper, Izmailov (Math Program Ser A 147:581–590, 2014) derived an extension of Robinson’s implicit function theorem for nonsmooth generalized equations in finite dimensions, which reduces to Clarke’s inverse function theorem when the generalized equation is just an equation. Páles (J Math Anal Appl 209:202–220, 1997) gave earlier a generalization of Clarke’s inverse function theorem to Banach spaces by employing Ioffe’s strict pre-derivative. In this paper we generalize both theorems of Izmailov and Páles to nonsmooth generalized equations in Banach spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Shortly before this paper was accepted for publication we received a letter by A. Izmailov where he confirmed that indeed his proof is not complete and could be fixed by using some of the arguments in our proof. He also noted that his proof heavily relies on the finite dimensions.
References
Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64, 97–102 (1976)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer, Dordrecht (2014)
Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc. 266, 1–56 (1981)
Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)
Izmailov, A.F.: Strongly regular nonsmooth generalized equations. Math. Program. Ser. A 147, 581–590 (2014)
Klatte, D., Kummer, B.: Nonsmooth equations in optimization. Regularity, calculus, methods and applications. In: Nonconvex Optimization and Its Applications, vol. 60. Kluwer, Dordrecht (2002)
Kummer, B.: An implicit-function theorem for \(C^{0,1}\)-equations and parametric \(C^{1,1}\)-optimization. J. Math. Anal. Appl. 158, 35–46 (1991)
Páles, Z.: Linear selections for set-valued functions and extensions of bilinear forms. Arch. Math. 62, 427–432 (1994)
Páles, Z.: Inverse and implicit function theorems for nonsmooth maps in Banach spaces. J. Math. Anal. Appl. 209, 202–220 (1997)
Páles, Z.: On abstract control problems with non-smooth data. In: Recent Advances in Optimization, pp. 205–216. Springer, Berlin (2006)
Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16, 292–309 (1991)
Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
A. L. Dontchev: Supported by the National Science Foundation Grant DMS 1008341.
Rights and permissions
About this article
Cite this article
Cibulka, R., Dontchev, A.L. A nonsmooth Robinson’s inverse function theorem in Banach spaces. Math. Program. 156, 257–270 (2016). https://doi.org/10.1007/s10107-015-0877-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-015-0877-2
Keywords
- Robinson’s inverse function theorem
- Clarke’s inverse function theorem
- Strict pre-derivative
- Generalized equation
- Strong metric regularity