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Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivatives

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Abstract

For a \(\mathcal {C}^2\)-smooth function on a finite-dimensional space, a necessary condition for its quasiconvexity is the positive semidefiniteness of its Hessian matrix on the subspace orthogonal to its gradient, whereas a sufficient condition for its strict pseudoconvexity is the positive definiteness of its Hessian matrix on the subspace orthogonal to its gradient. Our aim in this paper is to extend those conditions for \(\mathcal {C}^{1,1}\)-smooth functions by using the Fréchet and Mordukhovich second-order subdifferentials.

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References

  1. Aussel, D., Corvellec, J.N., Lassonde, M.: Subdifferential characterization of quasiconvexity and convexity. J. Convex Anal. 1, 195–201 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Aussel, D.: Subdifferential properties of quasiconvex and pseudoconvex functions: unified approach. J. Optim. Theory Appl. 97, 29–45 (1998)

    Article  MathSciNet  Google Scholar 

  3. Avriel, M.: r-Convex functions. Math. Program. 2, 309–323 (1972)

    Article  Google Scholar 

  4. Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity, Mathematical Concepts and Methods in Science and Engineering, vol. 36. Plenum Press, New York (1988)

    MATH  Google Scholar 

  5. Barron, E.N., Goebel, R., Jensen, R.R.: The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete Contin. Dyn. Syst. Ser. B 17, 1693–1706 (2012)

    Article  MathSciNet  Google Scholar 

  6. Barron, E.N., Goebel, R., Jensen, R.R.: Quasiconvex functions and nonlinear PDEs. Trans. Am. Math. Soc. 365, 4229–4255 (2013)

    Article  MathSciNet  Google Scholar 

  7. Cambina, A., Martein, L.: Generalized Convexity and Optimization. Theory and Applications. Springer, Berlin (2009)

    Google Scholar 

  8. Chieu, N.H., Lee, G.M., Yen, N.D.: Second-order subdifferentials and optimality conditions for \(C^1\)-smooth optimization problems. Appl. Anal. Optim. 1, 461–476 (2017)

    MathSciNet  Google Scholar 

  9. Crouzeix, J.-P.: On second order conditions for quasiconvexity. Math. Program. 18, 349–352 (1980)

    Article  MathSciNet  Google Scholar 

  10. Crouzeix, J.P., Ferland, J.A.: Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons. Math. Program. 23, 193–205 (1982)

    Article  MathSciNet  Google Scholar 

  11. Crouzeix J.-P.: Characterizations of generalized convexity and monotonicity, a survey. In: Generalized Convexity, Generalized Monotonicity, pp. 237 – 256 (1998)

  12. Ginchev, I., Ivanov, V.I.: Second-order characterizations of convex and pseudoconvex functions. J. Appl. Anal. 9, 261–273 (2003)

    Article  MathSciNet  Google Scholar 

  13. Hiriart-Urruty, J.B., Strodiot, J.J., Hien, N.V.: Generalized Hessian matrix and second order optimality conditions for problems with \(\cal{C}^{1,1}\) data. Appl. Math. Optim. 11, 43–56 (1984)

    Article  MathSciNet  Google Scholar 

  14. Hoa, B.T., Khanh, P.D., Trinh, T.T.T.: Characterizations of nonsmooth robustly quasiconvex functions. J. Optim. Theory Appl. 180, 775–786 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Luc, D.T.: Characterizations of quasiconvex function. Bull. Aust. Math. Soc. 48, 393–405 (1993)

    Article  MathSciNet  Google Scholar 

  17. Luc, D.T.: Taylor’s formula for \(C^{k,1}\) functions. SIAM J. Optim. 5, 659–669 (1995)

    Article  MathSciNet  Google Scholar 

  18. Mangasarian, O.L.: Nonlinear Programming. Classics in Applied Mathematics. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  19. Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55, 685–708 (2006)

    Article  MathSciNet  Google Scholar 

  20. Mordukhovich, B.S.: Variational analysis and generalized differentiation. In: Basic Theory, vol. I. Applications, vol. II. Springer, Berlin (2006)

  21. Nadi, M.T., Zafarani, J.: Characterizations of quasiconvex and pseudoconvex functions by their second-order regular subdifferentials. J. Aust. Math. Soc. (2019). https://doi.org/10.1017/S1446788719000090

    Article  MATH  Google Scholar 

  22. Soleimani-damaneh, M.: Characterization of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330, 1387–1392 (2007)

    Article  MathSciNet  Google Scholar 

  23. Penot, J.P., Quang, P.H.: On generalized convexity of functions and generalized monotonicity of set-valued maps. J. Optim. Theory Appl. 92, 343–356 (1997)

    Article  MathSciNet  Google Scholar 

  24. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  25. Yang, X.Q.: Continuous generalized convex functions and their characterizations. Optimization 54, 495–506 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editors and two anonymous referees for constructive comments and suggestions, which greatly improved the paper. Pham Duy Khanh was supported, in part, by the Fondecyt Postdoc Project 3180080, the Basal Program CMM–AFB 170001 from CONICYT–Chile, and the National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2017.325.

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Authors and Affiliations

  1. Department of Mathematics, HCMC University of Education, Ho Chi Minh, Vietnam

    Pham Duy Khanh

  2. Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile

    Pham Duy Khanh

  3. Department of Mathematics, HCMC University of Education, Ho Chi Minh, Vietnam

    Vo Thanh Phat

  4. Wayne State University, Detroit, MI, USA

    Vo Thanh Phat

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  1. Pham Duy Khanh
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Correspondence to Pham Duy Khanh.

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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). https://doi.org/10.1007/s11590-020-01563-6

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  • DOI: https://doi.org/10.1007/s11590-020-01563-6

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