Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions

Aram V. Arutyunov () and Sergey E. Zhukovskiy ()
Additional contact information
Aram V. Arutyunov: V. A. Trapeznikov Institute of Control Sciences of RAS
Sergey E. Zhukovskiy: V. A. Trapeznikov Institute of Control Sciences of RAS

Journal of Optimization Theory and Applications, 2023, vol. 199, issue 1, No 4, 112-142

Abstract: 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.

Keywords: Implicit function; Smoothing of locally Lipschitzian mapping; Generalized Jacobian; Mixed constraints; 26B10; 49J52; 34H05 (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-023-02244-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:199:y:2023:i:1:d:10.1007_s10957-023-02244-x

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-023-02244-x

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().