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
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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Journal of Mathematical Sciences (New York), 2015, 209:1, 66–87
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
\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.
\jour J. Math. Sci.
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This publication is cited in the following articles:
G. E. Ivanov, “Sharp estimates for the moduli of continuity of metric projections onto weakly convex sets”, Izv. Math., 79:4 (2015), 668–697
G. E. Ivanov, M. C. Lopushanski, “Teorema ob otdelimosti dlya nevypuklykh mnozhestv i eë prilozheniya”, Fundament. i prikl. matem., 21:4 (2016), 23–66
F. S. Stonyakin, “A Sublinear Analog of the Banach–Mazur Theorem in Separable Convex Cones with Norm”, Math. Notes, 104:1 (2018), 111–120
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