Abstract
In this paper we consider spaces with an asymmetric seminorm and continue to study weakly convex sets. If we consider the Minkowski functional of the epigraph of some convex function as a seminorm, then the results obtained for weakly convex sets can be applied to weakly convex functions whose epigraphs are weakly convex sets with respect to this seminorm. We consider two sets in an asymmetric seminormed space, one of which is weakly convex with respect to an asymmetric seminorm, and the other one is strongly convex with respect to the asymmetric seminorm. We study the nearest points (in the sense of seminorm) problem and prove that this problem is well posed. Well posedness is an important property in the optimization theory. If a minimization problem is well posed, then one can build stable numerical algorithms, used for finding the solution of the problem.
Keywords
Supported by the Russian Foundation for Basic Research, grant 18-01-00209.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Borodin, P.: On the convexity of \(n\)-chebyshev sets. Izv. RAN. Ser. Mat. 75(5), 19–46 (2011)
Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and lower–\(c^{2}\) property. J. Convex Anal. 2(1–2), 117–144 (1995)
Cobzas, S.: Separation of convex sets and best approximation in spaces with asymmetric norm. Quaestiones Math. 27(3), 275–296 (2004)
Cobzas, S.: Ekeland variational principle in asymmetric locally convex spaces. Topol. Appl. 159(10–11), 2558–2569 (2012)
Cobzas, S.: Functional Analysis in Asymmetric Normed Spaces. Birkhauser, Basel (2013)
Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
Ivanov, G.E., Golubev, M.O.: Alternative theorem for differential games with strongly convex admissible control sets. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds.) OPTIMA 2018. CCIS, vol. 974, pp. 321–335. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10934-9_23
Ivanov, G., Golubev, M.: Strong and weak convexity in nonlinear differential games. IFAC-PapersOnLine 51, 13–18 (2018)
Ivanov, G.E., Lopushanski, M.S., Golubev, M.O.: The nearest point theorem for weakly convex sets in asymmetric seminormed spaces. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds.) OPTIMA 2018. CCIS, vol. 974, pp. 21–34. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10934-9_2
Ivanov, G., Lopushanski, M.: Well-posedness of approximation and optimization problems for weakly convex sets and functions. J. Math. Sci. (United States) 209, 66–87 (2015)
Ivanov, G.E.: On well posed best approximation problems for a nonsymmetric seminorm. J. Convex Anal. 20(2), 501–529 (2013)
Ivanov, G.E.: Continuity and selections of the intersection operator applied to nonconvex sets. J. Convex Anal. 22(4), 939–962 (2015)
Ivanov, G.E.: Weak convexity of sets and functions in a banach space. J. Convex Anal. 22(2), 365–398 (2015)
Ivanov, G.E., Lopushanski, M.S.: Separation theorem for nonconvex sets and its applications. Fundam. Appl. Math. 21(4), 23–65 (2016)
Ivanov, G.E., Lopushanski, M.S.: Separation theorems for nonconvex sets in spaces with non-symmetric seminorm. J. Math. Inequalities Appl. 20(3), 737–754 (2017)
Ivanov, G.: Weak convexity of functions and the infimal convolution. J. Convex Anal. 23, 719–732 (2016)
Ivanov, G., Thibault, L.: Infimal convolution and optimal time control problem ii: limiting subdifferential. Set-Valued Variational Anal. 25(3), 517–542 (2017)
Ivanov, G., Thibault, L.: Infimal convolution and optimal time control problem i: Fréchet and proximal subdifferentials. Set-Valued Variational Anal. 26, 581–606 (2018)
Ivanov, G., Thibault, L.: Infimal convolution and optimal time control problem iii: minimal time projection set. SIAM J. Optim. 28(1), 30–44 (2018)
Ivanov, G., Thibault, L.: Well-posedness and subdifferentials of optimal value and infimal convolution. Set-Valued Variational Anal. 27, 841–861 (2019)
Jordan-Pérez, N., Sánchez-Perez, E.: Extreme points and geometric aspects of compact convex sets in asymmetric normed spaces. Topol. Appl. 203, 15–21 (2016)
Lopushanski, M.: Weakly convex sets in asymmetric normed spaces. In: Abstracts of the International Conference “Constructive Nonsmooth Analysis and Related Topics” Dedicated to the Memory of Professor V. F. Demyanov, pp. 34–38. Sankt-Petersburg (2017)
Lopushanski, M.S.: Normal regularity of weakly convex sets in asymmetric normed spaces. J. Convex Anal. 25(3), 737–758 (2018)
Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352, 5231–5249 (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Lopushanski, M.S. (2020). Well Posedness of the Nearest Points Problem for Two Sets in Asymmetric Seminormed Spaces. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2019. Communications in Computer and Information Science, vol 1145. Springer, Cham. https://doi.org/10.1007/978-3-030-38603-0_34
Download citation
DOI: https://doi.org/10.1007/978-3-030-38603-0_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38602-3
Online ISBN: 978-3-030-38603-0
eBook Packages: Computer ScienceComputer Science (R0)