Directional Metric Pseudo Subregularity of Set-valued Mappings: a General Model
Article
First Online:
- 7 Downloads
Abstract
This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established.
Keywords
Abstract subdifferential Metric regularity Directional metric regularity Metric subregularity Directional Hölder metric subregularity Directional metric pseudo-subregularity Coderivative SlopeMathematics Subject Classification (2010)
49J52 49J53 90C30Preview
Unable to display preview. Download preview PDF.
Notes
Acknowledgments
We gratefully acknowledge the referees for their constructive comments and suggestions.
References
- 1.Arutyunov, A.V., Avakov, E.R., Izmailov, A.F.: Directional regularity and metric regularity. SIAM J. Optim. 18(3), 810–833 (2007). https://doi.org/10.1137/060651616 MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Azé, D., Corvellec, J.N.: On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J. Optim. 12(4), 913–927 (2002)MathSciNetCrossRefGoogle Scholar
- 3.Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefGoogle Scholar
- 4.Borwein, J.M.: Stability and regular points of inequality systems. J. Optim. Theory Appl. 48(1), 9–52 (1986)MathSciNetCrossRefGoogle Scholar
- 5.Cibulka, R., Fabian, M., Kruger, A.: On semiregularity of mappings. J. Math. Anal. Appl. 473, 811–836 (2019)MathSciNetCrossRefGoogle Scholar
- 6.De Giorgi, E., Marino, A., Tosques, M.: Evolution problerns in metric spaces and steepest descent curves. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187 (1980). In Italian. English translation: Ennio De Giorgi, Selected Papers, Springer, Berlin 2006, 527–533zbMATHGoogle Scholar
- 7.Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2 Edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)Google Scholar
- 8.Facchinei, F., Fischer, A., Herrich, M.: An lp-newton method: nonsmooth equations, kkt systems, and nonisolated solutions. Math. Program. 146(1-2, Ser. A), 1–36 (2014)MathSciNetCrossRefGoogle Scholar
- 9.Facchinei, F., P.J.S.: Finite-dimensional variational inequalities and complementarity problems. In: Vol. II. Springer Series in Operations Research. Springer, New York (2003)Google Scholar
- 10.Gfrerer, H.: On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs. SIAM J. Optim. 23(1), 632–665 (2013). https://doi.org/10.1137/120891216 MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Gfrerer, H.: On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs. Set-Valued Var. Anal 22(1), 79–115 (2014). https://doi.org/10.1007/s11228-013-0266-z MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Huynh, V.N., Nguyen, H.T., Théra, M.: Directional Hölder metric regularity. J. Optim. Theory Appl. 171(3), 785–819 (2016)MathSciNetCrossRefGoogle Scholar
- 13.Huynh, V.N., 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 MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Huynh, V.N., Théra, M.: Directional metric regularity of multifunctions. Math. Oper. Res. 40(4), 969–991 (2015). https://doi.org/10.1287/moor.2014.0705 MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Ioffe, A.D.: On regularity concepts in variational analysis. J. Fixed Point Theory Appl. 8(2), 339–363 (2010). https://doi.org/10.1007/s11784-010-0021-0 MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Ioffe, A.D.: Convexity and variational analysis. In: Bailey, D.H., Bauschke, H.H., Borwein, P., Garvan, F., Théra, F., D. V.J., Wolkowicz, H. (eds.) Computational and Analytical Mathematics, Springer Proc. Math. Stat., vol. 50, pp 411–444. Springer, New York (2013), https://doi.org/10.1007/978-1-4614-7621-4-19
- 17.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 MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Ioffe, A.D.: Metric regularity – a survey. Part II. Applications. J. Aust. Math. Soc. 101(3), 376–417 (2016). https://doi.org/10.1017/S1446788715000695 MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Ioffe, A.D.: Variational analysis of regular mappings. Theory and applications. Springer monographs in mathematics springer (2017)CrossRefGoogle Scholar
- 20.Izmailov, A.F., Solodov, M.V.: Newton-type methods for optimization and variational problems. In: Springer Series in Operations Research and Financial Engineering, SpringerGoogle Scholar
- 21.Kruger, A. Y.: Error bounds and Hölder metric subregularity. Set-Valued Var. Anal. 23(4), 705–736 (2015)MathSciNetCrossRefGoogle Scholar
- 22.Kruger, A. Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015)MathSciNetCrossRefGoogle Scholar
- 23.Kruger, Alexander Y.: Nonlinear metric subregularity. J. Optim. Theory Appl. 171(3), 820–855 (2016). https://doi.org/10.1007/s10957-015-0807-8 MathSciNetCrossRefzbMATHGoogle Scholar
- 24.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)CrossRefGoogle Scholar
- 25.Ngai, H.V., Tinh, P.N.: Metric subregularity of multifunctions: First and second order infinitesimal characterizations. Math. Oper. Res. 40(3), 703–724 (2015). https://doi.org/10.1287/moor.2014.0691 MathSciNetCrossRefzbMATHGoogle Scholar
- 26.Ngai, H.V., Tron, N.H., Tinh, P.N.: Directional hölder metric subregularity and application to tangent cones. J. Convex Anal. 24(2), 417–457 (2017)MathSciNetzbMATHGoogle Scholar
- 27.Penot, J.P.: Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13(6), 629–643 (1989). https://doi.org/10.1016/0362-546X(89)90083-7 MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Penot, J.P.: Calculus without Derivatives Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)CrossRefGoogle Scholar
- 29.Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
Copyright information
© Springer Nature B.V. 2019