Coincidence Points of Parameterized Generalized Equations with Applications to Optimal Value Functions

The paper studies coincidence points of parameterized set-valued mappings (multifunctions), which provide an extended framework to cover several important topics in variational analysis and optimization that include the existence of solutions of parameterized generalized equations, implicit function...

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Bibliographic Details
Published inJournal of optimization theory and applications Vol. 196; no. 1; pp. 177 - 198
Main Authors Arutyunov, Aram V., Mordukhovich, Boris S., Zhukovskiy, Sergey E.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.01.2023
Springer Nature B.V
Subjects
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Continuity (mathematics)
Engineering
Existence theorems
Fixed points (mathematics)
Mathematical analysis
Mathematical functions
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Parameterization
Theory of Computation
Generalized equations
90C31
Variational analysis and generalized differentiation
Parametric optimization
47H10
49J53
49J52
Coincidence and implicit function theorems
Optimal value functions
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Summary:The paper studies coincidence points of parameterized set-valued mappings (multifunctions), which provide an extended framework to cover several important topics in variational analysis and optimization that include the existence of solutions of parameterized generalized equations, implicit function and fixed-point theorems, optimal value functions in parametric optimization, etc. Using the advanced machinery of variational analysis and generalized differentiation that furnishes complete characterizations of well-posedness properties of multifunctions, we establish a general theorem ensuring the existence of parameter-dependent coincidence point mappings with explicit error bounds for parameterized multifunctions between infinite-dimensional spaces. The obtained major result yields a new implicit function theorem and allows us to derive efficient conditions for semicontinuity and continuity of optimal value functions associated with parametric minimization problems subject to constraints governed by parameterized generalized equations.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-022-02140-w