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

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Abstract

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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Acknowledgements

The authors are grateful to two anonymous referees for their helpful remarks and suggestions, which allowed us to improve the original manuscript.

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Authors and Affiliations

  1. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Aram V. Arutyunov & Sergey E. Zhukovskiy

  2. Department of Mathematics, Wayne State University, Detroit, USA

    Boris S. Mordukhovich

Authors
  1. Aram V. Arutyunov
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  2. Boris S. Mordukhovich
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  3. Sergey E. Zhukovskiy
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Corresponding author

Correspondence to Boris S. Mordukhovich.

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Communicated by Michel Théra.

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Research of this author was partly supported by the Russian Science Foundation, project 22-21-00863.

Research of this author was partly supported by the USA National Science Foundation under grants DMS-1808978 and DMS-2204519, by the Australian Research Council under grant DP-190100555, and by Project 111 of China under grant D21024.

Research of this author was partly supported by the Russian Science Foundation, project 20-11-20131.

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Arutyunov, A.V., Mordukhovich, B.S. & Zhukovskiy, S.E. Coincidence Points of Parameterized Generalized Equations with Applications to Optimal Value Functions. J Optim Theory Appl 196, 177–198 (2023). https://doi.org/10.1007/s10957-022-02140-w

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  • DOI: https://doi.org/10.1007/s10957-022-02140-w

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