Equations on monotone graphs

This paper studies the local analysis of equations on a product U ×  U of Banach spaces, whose variables lie in a subset having the special property that it is locally Lipschitz-homeomorphic to an open subset of U . A prominent example, to which we devote most of the paper, is a system of equations...

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Veröffentlicht in:Mathematical programming Jg. 141; H. 1-2; S. 49 - 101
1. Verfasser: Robinson, Stephen M.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2013
Springer Nature B.V
Schlagworte:
Banach space
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Feasibility
Full Length Paper
Graphs
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinear programming
Numerical Analysis
Operators
Representations
Studies
Theoretical
Vector space
United States > US
90C31
Variational condition
Normal manifold
Normal map
Complementarity
49J53
Variational inequality
49K40
49J40
90C33
Monotone graph
Implicit function
ISSN:0025-5610, 1436-4646
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Beschreibung
Zusammenfassung:This paper studies the local analysis of equations on a product U ×  U of Banach spaces, whose variables lie in a subset having the special property that it is locally Lipschitz-homeomorphic to an open subset of U . A prominent example, to which we devote most of the paper, is a system of equations whose variables lie in the graph of a maximal monotone operator. This general formulation covers many specific problems of interest, and our objective is to understand the local behavior of solutions of such equations when they depend on parameters. We analyze this local behavior in stages, the first stage being to apply a tailored implicit-function theorem to an abstract formulation of the problem, thereby producing a set of results applicable to any particular problem instance. The second stage is to specialize the analysis to a Hilbert space H , with the subset mentioned above being the graph of a maximal monotone operator on H . This makes the results of the first stage applicable to many variational problems of practical importance. We then develop in detail the analytical steps to apply these results to finite-dimensional variational conditions with constraints of generalized nonlinear-programming type. The conditions thus identified generalize the strong second-order sufficient condition and linear-independence constraint qualification of nonlinear programming. A detailed example brings out some of the issues involved in practical implementation of this method. It also shows that aspects of representation (problem formulation) can strongly influence the feasibility of local analysis. This sensitivity to representation does not seem to be well known.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-011-0509-4