This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aragón Artacho, F.J., Mordukhovich, B.S.: Enhanced metric regularity and Lipschitzian properties of variational systems. J. Glob. Optim. 50, 145–167 (2011)
Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka–Łojasiewicz inequality. Math. Oper. Res. 35, 438–457 (2010)
Bakan, A., Deutsch, F., Li, W.: Strong CHIP, normality, and linear regularity of convex sets. Trans. Amer. Math. Soc. 357, 3831–3863 (2005)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. Ser. A 86, 135–160 (1999)
Bauschke, H.H., Borwein, J.M., Tseng, P.: Bounded linear regularity, strong CHIP, and CHIP are distinct properties. J. Convex Anal. 7, 395–412 (2000)
Bui, H.T., Kruger, A.Y.: Extremality, stationarity and generalized separation of collections of sets. J. Optim. Theory Appl. 182, 211–264 (2019)
Cibulka, R., Fabian, M., Kruger, A.Y.: On semiregularity of mappings. J. Math. Anal. Appl. 473, 811–836 (2019)
Cuong, N.D., Kruger, A.Y.: Dual sufficient characterizations of transversality properties. Positivity (2020)
Cuong, N.D., Kruger, A.Y.: Nonlinear transversality of collections of sets: Primal space sufficient characterizations. arXiv:1902.06186 (2019)
Cuong, N.D., Kruger, A.Y.: Primal space necessary characterizations of transversality properties. Optimization Online 2020-01-7579 (2020)
Cuong, N.D., Kruger, A.Y.: Nonlinear transversality of collections of sets: Dual space necessary characterizations. J. Convex Anal. 27, 287–308 (2020)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)
Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15, 1637–1651 (2015)
Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)
Ioffe, A.D.: Variational Analysis of Regular Mappings. Theory and Applications. Springer Monographs in Mathematics. Springer International Publishing (2017)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Nonconvex Optimization and its Applications, vol. 60. Kluwer Academic Publishers, Dordrecht (2002)
Kruger, A.Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1, 101–126 (2005)
Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14, 187–206 (2006)
Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13, 1737–1785 (2009)
Kruger, A.Y.: About intrinsic transversality of pairs of sets. Set-Valued Var. Anal. 26, 111–142 (2018)
Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. J. Optim. Theory Appl. 154, 339–369 (2012)
Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization. J. Optim. Theory Appl. 155, 390–416 (2012)
Kruger, A.Y., Luke, D.R., Thao, N.H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25, 701–729 (2017)
Kruger, A.Y., Luke, D.R., Thao, N.H.: Set regularities and feasibility problems. Math. Program. Ser. B 168, 279–311 (2018)
Kruger, A.Y., Thao, N.H.: About uniform regularity of collections of sets. Serdica Math. J. 39, 287–312 (2013)
Kruger, A.Y., Thao, N.H.: About [q]-regularity properties of collections of sets. J. Math. Anal. Appl. 416, 471–496 (2014)
Kruger, A.Y., Thao, N.H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164, 41–67 (2015)
Kruger, A.Y., Thao, N.H.: Regularity of collections of sets and convergence of inexact alternating projections. J. Convex Anal. 23, 823–847 (2016)
Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9, 485–513 (2009)
Li, C., Ng, K.F., Pong, T.K.: The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 18, 643–665 (2007)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)
Ng, K.F., Yang, W.H.: Regularities and their relations to error bounds. Math. Program. Ser. A 99, 521–538 (2004)
Ng, K.F., Zang, R.: Linear regularity and ϕ-regularity of nonconvex sets. J. Math. Anal. Appl. 328, 257–280 (2007)
Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9, 187–216 (2001)
Noll, D., Rondepierre, A.: On local convergence of the method of alternating projections. Found. Comput. Math. 16, 425–455 (2016)
Penot, J.P.: Calculus Without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)
Zheng, X.Y., Ng, K.F.: Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19, 62–76 (2008)
Zheng, X.Y., Wei, Z., Yao, J.-C.: Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets. Nonlinear Anal. 73, 413–430 (2010)
The authors thank the referees for the careful reading of the manuscript and their constructive comments and suggestions. We are also grateful to editor Prof. Michel Théra for organizing the refereeing process perfectly.
The research was supported by the Australian Research Council, project DP160100854. The first author is supported by an Australian Government Research Training Program (RTP) Stipend and RTP Fee-Offset Scholarship through Federation University Australia. The third author benefited from the support of the FMJH Program PGMO and Conicyt REDES program 180032.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dedicated to Professor Marco Antonio López Cerdá on the occasion of his 70th birthday.
About this article
Cite this article
Bui, H.T., Cuong, N.D. & Kruger, A.Y. Geometric and Metric Characterizations of Transversality Properties. Vietnam J. Math. 48, 277–297 (2020). https://doi.org/10.1007/s10013-020-00388-1
Mathematics Subject Classification (2010)