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Second-order characterization of convex mappings in Banach spaces and its applications

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Abstract

We show that the positive semi-definiteness of the regular or limiting (Mordukhovich) second-order subdifferential of an approximately convex function is a sufficient condition for its convexity. As a consequence of our result, we obtain a second-order characterization for the class of lower-\(C^1\) functions. Furthermore, we show by an example that positive semi-definiteness of the second-order subdifferential of convex functions is not a necessary condition for some cases. Also, a second-order characterization for C-convex mappings is obtained, and derive some applications in optimization.

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References

  1. Atouch, H., Brézis, H.: Duality for the sum of convex functions. In: Barroso, J. A. Aspects of Mathematics and its Applications (eds.) Elsevier, Amsterdam (1986)

  2. Aussel, D., Corvellec, J.N., Lassonde, M.: Mean value property and subdifferential criteria for lower semicontinuous functions. Trans. Amer. Math. Soc. 347(10), 4147–4161 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357(4), 1275–1301 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bednarik, D., Pastor, K.: Elimination of strict convergence in optimization. SIAM J. Control. Optim. 43(3), 1063–1077 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  6. Chen, G., Huang, X., Yang, X.: Vector Optimization Set-valued and Variational Analysis. Lecture Notes in Economic and Mathematical Systems. Springer-verlag, Berlin (2005)

    MATH  Google Scholar 

  7. Chieu, N.H., Chuong, T.D., Yao, J.C., Yen, N.D.: Characterizing convexity of a function by its Fréchet and limiting second-order subdifferentials. Set-Valued Var. Anal. 19(1), 75–96 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chieu, N.H., Huy, N.Q.: Second-order subdifferentials and convexity of real-valued functions. Nonlinear Anal. 74(1), 154–160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chieu, N.H., Lee, G.M., Mordukhovich, B.S., Nghia, T.T.A.: Coderivative characterizations of maximal monotonicity for set-valued mappings. J. Convex Anal. 23(2), 461–480 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordecht (1990)

    Book  MATH  Google Scholar 

  11. Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control. Optim. 28, 789–809 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daniilidis, A., Georgiev, P.: Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291(1), 292–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gallier, J.: Basics of algebra, topology, and differential calculus. CiteSeerX (2013)

  14. Hiriart-Urruty, J.B.: Characterization of the plenary hull of the generalized Jacobian matrix. Math. Prog. Study. 17, 1–12 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ivanov, V.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Global Opt. 57(3), 677–693 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khanh, P.D., Phat, V.T.: Second-order characterizations of \(C^1\)-smooth robustly quasiconvex functions. Oper. Res. Lett. 46, 568–572 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khanh, P.D., Phat, V.T.: Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivatives. Optim. Lett. 14, 2413–2427 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. 116, 3325–3358 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lang, S.: Fundamentals of differential geometry. no. 191. In: Graduate Texts in Mathematics. Springer : New York(1999)

  20. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economic and Mathematical Systems. Springer-verlag, Berlin (1989)

    Book  Google Scholar 

  21. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control. Optim. 15, 957–972 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mordukhovich, B. S.:Sensitivity analysis in nonsmooth optimization. In: Field, D. A., Komkov, V. (eds.)Theoretical Aspects of Industrial Design, SIAM Proceedings in Applied Mathematics, pp. 32-42. SIAM Publications, Philadelphia, PA, 58 (1992)

  23. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, New York (2018)

    Book  MATH  Google Scholar 

  24. Mordukhovich, B.S.: Variational Analysis and Generalized Differential I II. Springer, New York (2006)

    Google Scholar 

  25. Mordukhovich, B.S., Nghia, T.T.A.: Local monotonicity and full stability for parametric variational systems. SIAM J. Optim. 26, 1032–1059 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23(3), 1810–1849 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nadi, M.T., Yao, J.C., Zafarani, J.: Second-order characterization of convex functions and its applications. J. Appl. Anal. 25(1), 49–58 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nadi, M.T., Zafarani, J.: Characterizations of quasiconvex and pseudoconvex functions by their second-order regular subdifferentials. J. Aust. Math. Soc. 109(2), 217–229 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nadi, M.T., Zafarani, J.: Second-order optimality conditions for constrained optimization problems with \(C^1\) data via regular and limiting subdifferentials. J. Optim. Theory Appl. 193, 158–179 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ngai, H.V., Luc, D.T., Thera, M.: Approximate convex functions. J. Nonlinear Convex Anal. 1(2), 155–176 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Nghia, T.T.A., Pham, D.T., Tran, T.T.T.: On the positive definiteness of limiting coderivative for set-valued mappings. Set-Valued Var. Anal (2020). https://doi.org/10.1007/s11228-020-00547-z

    Article  MATH  Google Scholar 

  32. Ning, E., Song, W., Zhang, Y.: Second-order sufficient optimality conditions in vector optimization. J. Glob. Optim. 54, 537–549 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8(2), 287–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rockafellar, R.T.: Favorable classes of Lipschitz continuous functions in subgradient optimization. In: Nurminski, E. (ed.) Nondifferentiable Optimization. Pergamon Press, New York (1982)

    Google Scholar 

  35. Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  36. Spingarn, J.E.: Submonotone subdifferentials of Lipschitz functions. Trans. Amer. Math. Soc. 264, 77–89 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  37. Taa, A.: Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Global Optim. 50(2), 271–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhu, Q.J.: The equivalence of several basic theorems for subdifferentials. Set-Valued Anal. 6, 171–185 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was partly supported by the Iranian National Science Foundation (INSF) under the contract No. 99014611. The authors are grateful to the anonymous reviewer for the careful reading of the manuscript and especially for the constructive comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

    Mohammad Taghi Nadi

  2. Department of Mathematics, Sheikhbahaee University and University of Isfahan, Isfahan, Iran

    Jafar Zafarani

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  1. Mohammad Taghi Nadi
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  2. Jafar Zafarani
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Correspondence to Jafar Zafarani.

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Nadi, M.T., Zafarani, J. Second-order characterization of convex mappings in Banach spaces and its applications. J Glob Optim 86, 1005–1023 (2023). https://doi.org/10.1007/s10898-023-01301-z

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