Robinson’s implicit function theorem and its extensions

S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the first-order optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the...

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Bibliographic Details
Published in:Mathematical programming Vol. 117; no. 1-2; pp. 129 - 147
Main Authors: Dontchev, A. L., Rockafellar, R. T.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.03.2009
Springer Nature B.V
Subjects:
Approximation
Banach spaces
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Localization
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Neighborhoods
Numerical Analysis
Optimization
Studies
Theoretical
90C31
49J53
Semiderivatives
47J07
Lipschitz modulus
Calmness
First-order approximations
Inverse and implicit function theorems
Variational inequalities
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the first-order optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the classical implicit function theorem, if not quite all, but went far beyond that in ideas and format. Here, Robinson’s theorem is viewed from the perspective of more recent developments in variational analysis as well as some lesser-known results in the implicit function literature on equations, prior to the advent of generalized equations. Extensions are presented which fully cover such results, translating them at the same time to generalized equations broader than variational inequalities. Robinson’s notion of first-order approximations in the absence of differentiability is utilized in part, but even looser forms of approximation are shown to furnish significant information about solutions.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-007-0161-1