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

This article is cited in 3 scientific papers (total in 3 papers)

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.

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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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  mathscinet
    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  mathnet  crossref  crossref  isi  elib
  • Фундаментальная и прикладная математика
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