RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Journal history Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2013, Volume 18, Issue 5, Pages 89–118 (Mi fpm1543)

Well-posedness of approximation and optimization problems for weakly convex sets and functions

G. E. Ivanov, M. S. Lopushanski

Moscow Institute of Physics and Technology (State University), Moscow, Russia

Abstract: We consider the class of weakly convex sets with respect to a quasiball in a Banach space. This class generalizes the classes of sets with positive reach, proximal smooth sets and prox-regular sets. We prove the well-posedness of the closest points problem of two sets, one of which is weakly convex with respect to a quasiball $M$, and the other one is a summand of the quasiball $-rM$, where $r\in(0,1)$. We show that if a quasiball $B$ is a summand of a quasiball $M$, then a set that is weakly convex with respect to the quasiball $M$ is also weakly convex with respect to the quasiball $B$. We consider the class of weakly convex functions with respect to a given convex continuous function $\gamma$ that consists of functions whose epigraphs are weakly convex sets with respect to the epigraph of $\gamma$. We obtain a sufficient condition for the well-posedness of the infimal convolution problem, and also a sufficient condition for the existence, uniqueness, and continuous dependence on parameters of the minimizer.

Full text: PDF file (281 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences (New York), 2015, 209:1, 66–87

Bibliographic databases:

UDC: 517.982.252

Citation: G. E. Ivanov, M. S. Lopushanski, “Well-posedness of approximation and optimization problems for weakly convex sets and functions”, Fundam. Prikl. Mat., 18:5 (2013), 89–118; J. Math. Sci., 209:1 (2015), 66–87

Citation in format AMSBIB
\Bibitem{IvaLop13} \by G.~E.~Ivanov, M.~S.~Lopushanski \paper Well-posedness of approximation and optimization problems for weakly convex sets and functions \jour Fundam. Prikl. Mat. \yr 2013 \vol 18 \issue 5 \pages 89--118 \mathnet{http://mi.mathnet.ru/fpm1543} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431846} \transl \jour J. Math. Sci. \yr 2015 \vol 209 \issue 1 \pages 66--87 \crossref{https://doi.org/10.1007/s10958-015-2485-3} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84938292380} 

• http://mi.mathnet.ru/eng/fpm1543
• http://mi.mathnet.ru/eng/fpm/v18/i5/p89

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. E. Ivanov, “Sharp estimates for the moduli of continuity of metric projections onto weakly convex sets”, Izv. Math., 79:4 (2015), 668–697
2. G. E. Ivanov, M. C. Lopushanski, “Teorema ob otdelimosti dlya nevypuklykh mnozhestv i eë prilozheniya”, Fundament. i prikl. matem., 21:4 (2016), 23–66
3. F. S. Stonyakin, “A Sublinear Analog of the Banach–Mazur Theorem in Separable Convex Cones with Norm”, Math. Notes, 104:1 (2018), 111–120
•  Number of views: This page: 229 Full text: 94 References: 35