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 ∈ (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 γ that consists of functions whose epigraphs are weakly convex sets with respect to the epigraph of γ. 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.
Similar content being viewed by others
References
A. R. Alimov and M. I. Karlov, “Sets with external Chebyshev layer,” Math. Notes, 69, No. 2, 269–273 (2001).
M. V. Balashov and G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces,” Izv. Math., 73, 455–499 (2009).
M. V. Balashov and E. S. Polovinkin, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow (2004).
M. V. Balashov and D. Repovš, “Uniform convexity and the splitting problem for selections,” J. Math. Anal. Appl., 360, No. 1, 307–316 (2009).
F. Bernard, L. Thibault, and N. Zlateva, “Characterization of proximal regular sets in super reflexive Banach spaces,” J. Convex Anal., 13, 525–559 (2006).
F. Bernard, L. Thibault, and N. Zlateva, “Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties,” Trans. Am. Math. Soc., 363, 2211–2247 (2011).
F. H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and lower-C 2 property,” J. Convex Anal., 2, 117–144 (1995).
J. A. Clarkson, “Uniformly convex spaces,” Trans. Am. Math. Soc., 40, 396–414 (1936).
G. Colombo, V. V. Goncharov, and B. S. Mordukhovich, “Well-posedness of minimal time problems with constant dynamics in Banach spaces,” Set-Valued Var. Anal., 18, No. 3-4, 349–372 (2010).
H. Federer, “Curvature measures,” Trans. Am. Math. Soc., 93, 418–491 (1959).
W. Fenchel, Convex Cones, Sets and Functions, Mimeographed Lect. Notes, Princeton Univ. (1951).
V. V. Goncharov and F. F. Pereira, “Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals,” J. Convex Anal., 18, 1–36 (2011).
V. V. Goncharov and F. F. Pereira, “Geometric conditions for regularity in a time-minimum problem with constant dynamics,” J. Convex Anal., 19, 631–669 (2012).
G. E. Ivanov, “A criterion for smooth generating sets,” Sb. Math., 198, No. 3, 343–368 (2007).
G. E. Ivanov, “On well posed best approximation problems for a nonsymmetric seminorm,” J. Convex Anal., 20, No. 2, 501–529 (2013).
G. E. Ivanov, “Weak convexity of sets and functions in a Banach space,” J. Convex Anal., to appear.
G. E. Ivanov, Weakly Convex Sets and Functions. Theory and Applications [in Russian], Fizmatlit, Moscow (2006).
G. E. Ivanov and M. S. Lopushanski, “Approximate properties of weakly convex sets in spaces with nonsymmetric seminorm,” Tr. MFTI, 4, No. 4, 94–104 (2012).
R. Janin, Sur la dualité et la sensibilité dans les problèmes de programmation mathématique, Thèse de Doctorat ès-Sciences Mathématiques, Université de Paris (1974).
E. S. Levitin and B. T. Polyak, “Constrained minimization methods,” USSR Comput. Math. Math. Phys., 6, No. 5, 1–50 (1966).
J. J. Moreau, “Proximité et dualité dans un espace hilbertien,” Bull. Soc. Math. Fr., 93, 273–299 (1965).
R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Trans. Am. Math. Soc., 352, 5231–5249 (2000).
R. T. Rockafellar, Convex Analysis, Princeton, New Jersey (1970).
R. T. Rockafellar, “Favorable classes of Lipschitz continuous functions in subgradient optimization,” in: E. Nurminski (ed.), Progress in Nondifferentiable Optimization, IIASA Collab. Proc. Ser., Int. Inst. Appl. Systems Anal., 125–144 (1982).
J.-P. Vial, “Strong and weak convexity of sets and functions,” Math. Oper. Res., 8, 231–259 (1983).
K. Yosida, Functional Analysis, Springer, Berlin (1964).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 5, pp. 89–118, 2013.
Rights and permissions
About this article
Cite this article
Ivanov, G.E., Lopushanski, M.S. Well-Posedness of Approximation and Optimization Problems for Weakly Convex Sets and Functions. J Math Sci 209, 66–87 (2015). https://doi.org/10.1007/s10958-015-2485-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2485-3