Skip to main content
SpringerLink
Log in

On implicit function theorem for locally Lipschitz equations

  • Short Communication
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

Equations defined by locally Lipschitz continuous mappings with a parameter are considered. Implicit function theorems for this equation are obtained. The regularity condition is formulated in the terms of the Clarke Jacobian. Implicit functions estimates are derived. It is shown that the considered regularity assumptions are weaker than most of the known ones. The obtained implicit function theorems are applied to derive conditions for upper semicontinuity of the optimal value function for parameterized optimization problems

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Arutyunov, A.V., Izmailov, A.F.: Sensitivity analysis for cone-constrained optimization problems under the relaxed constraint qualifications. Math. Oper. Res. 30(2), 333–353 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arutyunov, A.V., Izmailov, A.F., Zhukovskiy, S.E.: Continuous selections of solutions for locally Lipschitzian equations. J. Optim. Theory Appl. 185, 679–699 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arutyunov, A.V., Zhukovskiy, S.E.: Existence and properties of inverse mappings. Proc. Steklov Inst. Math. 271, 12–22 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arutyunov, A.V., Zhukovskiy, S.E.: Stable solvability of nonlinear equations under completely continuous perturbations. Proc. Steklov Inst. Math. 312, 1–15 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Akademie-Verlag, Berlin (1982)

    Book  MATH  Google Scholar 

  6. Bartl, D., Fabian, M.: Can Pourciau’s open mapping theorem be derived from Clarke’s inverse mapping theorem easily. J. Math. Anal. Appl. 497(2), 124858 (2021)

  7. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  8. Cibulka, R., Fabian, M.: Continuous selections for inverse mappings in Banach spaces. J. Math. Anal. Appl. 499(1), 125025 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64(1), 97–102 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  11. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer, Dordrecht (2014)

    Book  MATH  Google Scholar 

  12. Izmailov, A.F.: Strongly regular nonsmooth generalized equations. Math. Program. Ser. A 147, 581–590 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  14. Klatte, D.: On the lower semicontinuity of optimal sets in convex parametric optimization. In: Point-to-Set Maps and Mathematical Programming, pp. 104–109. Springer, Berlin (1979)

  15. Kummer, B.: Stability and weak duality in convex programming without regularity. In: Wissenschaftliche Zeitschrift der Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Reihe XXX, pp. 381–386 (1981)

  16. Magaril-Il’aev, G.G.: The implicit function theorem for Lipschitz maps. Russ. Math. Surv. 33(1), 209–210 (1978)

  17. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988)

    MATH  Google Scholar 

  18. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, New York (2006)

    Book  Google Scholar 

  19. Pourciau, B.H.: Analysis and optimization of Lipschitz continuous mappings. J. Optim. Theory Appl. 22, 311–351 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rockafellar, R.T.: Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Program. Stud. 17, 28–66 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rockafellar, R.T.: Directional differentiability of the optimal value function in a nonlinear programming problem. Math. Program. Stud. 21, 213–226 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vinter, R.: Optimal Control. Birkhauser, Basel (2010)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Aram V. Arutyunov & Sergey E. Zhukovskiy

Authors
  1. Aram V. Arutyunov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey E. Zhukovskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research is supported by the grant of the President of Russian Federation (Project No. MD-2658.2021.1.1). Theorems 1 and 2 were obtained by the first author and supported by the Russian Science Foundation (Project No. 22-21-00863) The results in Sect. 4 were obtained by the second author and supported by the Russian Science Foundation (Project No. 20-11-20131)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arutyunov, A.V., Zhukovskiy, S.E. On implicit function theorem for locally Lipschitz equations. Math. Program. 198, 1107–1120 (2023). https://doi.org/10.1007/s10107-021-01750-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-021-01750-y

Keywords

Mathematics Subject Classification

Search

Navigation