Abstract
This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued map**s as well as two nonlinear transversality properties of a convex set-valued map** to a convex set in the range space are discussed.
Similar content being viewed by others
References
Bakan, A., Deutsch, F., Li, W.: Strong CHIP, normality, and linear regularity of convex sets. Trans. Am. Math. Soc. 357(10), 3831–3863 (2005). https://doi.org/10.1090/S0002-9947-05-03945-0
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996). https://doi.org/10.1137/S0036144593251710
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. Progr., Ser. A 86(1), 135–160 (1999). https://doi.org/10.1007/s101070050083
Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Progr., Ser. A 165(2), 471–507 (2017). https://doi.org/10.1007/s10107-016-1091-6
Borwein, J.M., Li, G., Tam, M.K.: Convergence rate analysis for averaged fixed point iterations in common fixed point problems. SIAM J. Optim. 27(1), 1–33 (2017). https://doi.org/10.1137/15M1045223
Borwein, J.M., Li, G., Yao, L.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets. SIAM J. Optim. 24(1), 498–527 (2014). https://doi.org/10.1137/130919052
Bui, H.T., Cuong, N.D., Kruger, A.Y.: Transversality of collections of sets: geometric and metric characterizations. Vietnam J. Math. 48(2), 277–297 (2020). https://doi.org/10.1007/s10013-020-00388-1
Chuong, T.D.: Metric regularity of a positive order for generalized equations. Appl. Anal. 94(6), 1270–1287 (2015). https://doi.org/10.1080/00036811.2014.930821
Chuong, T.D.: Stability of implicit multifunctions via point-based criteria and applications. J. Optim. Theory Appl. 183(3), 920–943 (2019). https://doi.org/10.1007/s10957-019-01562-3
Cibulka, R., Fabian, M., Kruger, A.Y.: On semiregularity of map**s. J. Math. Anal. Appl. 473(2), 811–836 (2019). https://doi.org/10.1016/j.jmaa.2018.12.071
Cuong, N.D., Kruger, A.Y.: Dual sufficient characterizations of transversality properties. Positivity (2020). https://doi.org/10.1007/s11117-019-00734-9
Cuong, N.D., Kruger, A.Y.: Nonlinear transversality of collections of sets: dual space necessary characterizations. J. Convex Anal. 27(1), 287–308 (2020)
Cuong, N.D., Kruger, A.Y.: Transversality properties: primal sufficient conditions. Set-Valued Var. Anal. (2020). https://doi.org/10.1007/s11228-020-00545-1
Dao, M.N., Phan, H.M.: Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems. J. Global Optim. 72(3), 443–474 (2018). https://doi.org/10.1007/s10898-018-0654-x
Dao, M.N., Phan, H.M.: Linear convergence of projection algorithms. Math. Oper. Res. 44(2), 715–738 (2019). https://doi.org/10.1287/moor.2018.0942
De Giorgi, E., Marino, A., Tosques, M.: Evolution problerns in in metric spaces and steepest descent curves. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187. In: Italian. English translation: Ennio De Giorgi, Selected Papers. Springer, Berlin 2006, pp. 527–533 (1980)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Map**s. A View from Variational Analysis, 2 edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014). https://doi.org/10.1007/978-1-4939-1037-3
Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15(6), 1637–1651 (2015). https://doi.org/10.1007/s10208-015-9279-3
Drusvyatskiy, D., Li, G., Wolkowicz, H.: A note on alternating projections for ill-posed semidefinite feasibility problems. Math. Progr., Ser. A 162(1–2), 537–548 (2017). https://doi.org/10.1007/s10107-016-1048-9
Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set-Valued Var. Anal. 18(2), 121–149 (2010)
Hesse, R., Luke, D.R.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23(4), 2397–2419 (2013). https://doi.org/10.1137/120902653
Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000). https://doi.org/10.1070/rm2000v055n03ABEH000292
Ioffe, A.D.: Metric regularity—a survey. Part I. Theory J. Aust. Math. Soc. 101(2), 188–243 (2016). https://doi.org/10.1017/S1446788715000701
Ioffe, A.D.: Variational Analysis of Regular Map**s. Theory and Applications. Springer Monographs in Mathematics. Springer (2017). https://doi.org/10.1007/978-3-319-64277-2
Kruger, A.Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1(1), 101–126 (2005)
Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14(2), 187–206 (2006). https://doi.org/10.1007/s11228-006-0014-8
Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13(6A), 1737–1785 (2009). https://doi.org/10.11650/twjm/1500405612
Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015). https://doi.org/10.1080/02331934.2014.938074
Kruger, A.Y.: About intrinsic transversality of pairs of sets. Set-Valued Var. Anal. 26(1), 111–142 (2018). https://doi.org/10.1007/s11228-017-0446-3
Kruger, A.Y., Luke, D.R., Thao, N.H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25(4), 701–729 (2017). https://doi.org/10.1007/s11228-017-0436-5
Kruger, A.Y., Luke, D.R., Thao, N.H.: Set regularities and feasibility problems. Math. Progr., Ser. B 168(1–2), 279–311 (2018). https://doi.org/10.1007/s10107-016-1039-x
Kruger, A.Y., Thao, N.H.: About \([q]\)-regularity properties of collections of sets. J. Math. Anal. Appl. 416(2), 471–496 (2014). https://doi.org/10.1016/j.jmaa.2014.02.028
Kruger, A.Y., Thao, N.H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164(1), 41–67 (2015). https://doi.org/10.1007/s10957-014-0556-0
Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9(4), 485–513 (2009). https://doi.org/10.1007/s10208-008-9036-y
Li, G.: Global error bounds for piecewise convex polynomials. Math. Progr. 137(1–2, Ser. A), 37–64 (2013). https://doi.org/10.1007/s10107-011-0481-z
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)
Ng, K.F., Zang, R.: Linear regularity and \(\phi \)-regularity of nonconvex sets. J. Math. Anal. Appl. 328(1), 257–280 (2007). https://doi.org/10.1016/j.jmaa.2006.05.028
Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9(1–2), 187–216 (2001). https://doi.org/10.1023/A:1011291608129
Ngai, H.V., Théra, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19(1), 1–20 (2008). https://doi.org/10.1137/060675721
Noll, D., Rondepierre, A.: On local convergence of the method of alternating projections. Found. Comput. Math. 16(2), 425–455 (2016). https://doi.org/10.1007/s10208-015-9253-0
Penot, J.P.: Calculus Without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-4538-8
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(1), 62–76 (2008). https://doi.org/10.1137/060659132
Zheng, X.Y., Wei, Z., Yao, J.C.: Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets. Nonlinear Anal. 73(2), 413–430 (2010). https://doi.org/10.1016/j.na.2010.03.032
Acknowledgements
We would like to thank the referees for the careful reading of the manuscript and their constructive comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research was supported by the Australian Research Council, Project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.
Rights and permissions
About this article
Cite this article
Cuong, N.D., Kruger, A.Y. Primal necessary characterizations of transversality properties. Positivity 25, 531–558 (2021). https://doi.org/10.1007/s11117-020-00775-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-020-00775-5