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Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions

Author

Listed:
  • Aram V. Arutyunov

    (V. A. Trapeznikov Institute of Control Sciences of RAS)

  • Sergey E. Zhukovskiy

    (V. A. Trapeznikov Institute of Control Sciences of RAS)

Abstract

The paper is devoted to the question of existence of continuous implicit functions for non-smooth equations. The equation $$f(x,\sigma )=0$$ f ( x , σ ) = 0 is studied with the unknown x and the parameter $$\sigma $$ σ . We assume that the mapping which defines the equation is locally Lipschitzian in the unknown variable, the parameter belongs to a topological space, and both the unknown variable and the value of the mapping belong to finite-dimensional spaces. Sufficient conditions for the existence of a continuous implicit function in a neighbourhood of a given value of the parameter and conditions for the existence of a continuous implicit function on a given subset of the parameter space are obtained in terms of the Clarke generalized Jacobian. The key tool of this development is the smoothing of the initial equation and application of recent results on solvability of smooth nonlinear equations and a priori solution estimates. Applications to control and optimization are discussed.

Suggested Citation

  • Aram V. Arutyunov & Sergey E. Zhukovskiy, 2023. "Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 112-142, October.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:1:d:10.1007_s10957-023-02244-x
    DOI: 10.1007/s10957-023-02244-x
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    References listed on IDEAS

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    1. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    2. Boris S. Mordukhovich & Bingwu Wang, 2004. "Restrictive metric regularity and generalized differential calculus in Banach spaces," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-28, January.
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    4. Aram V. Arutyunov & Alexey F. Izmailov & Sergey E. Zhukovskiy, 2020. "Continuous Selections of Solutions for Locally Lipschitzian Equations," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 679-699, June.
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