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CHARACTERIZATIONS OF QUASICONVEX AND PSEUDOCONVEX FUNCTIONS BY THEIR SECOND-ORDER REGULAR SUBDIFFERENTIALS

Part of: Functions of several variables Existence theories

Published online by Cambridge University Press:  30 April 2019

MOHAMMAD TAGHI NADI  and
JAFAR ZAFARANI Open the ORCID record for JAFAR ZAFARANI [Opens in a new window]
MOHAMMAD TAGHI NADI
Affiliation:
University of Isfahan, Isfahan, 81745-163, Iran email mt_nadi@yahoo.com
JAFAR ZAFARANI*
Affiliation:
University of Isfahan and Sheikhbahaee University, Isfahan, 81745-163, Iran email jzaf@zafarani.ir
*
✉jzaf@zafarani.ir
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Abstract

We present the second-order necessary and sufficient conditions for quasiconvex and pseudoconvex functions in terms of their second-order regular subdifferentials.

Keywords

quasiconvexitysecond-order subdifferentialpositive semidefinite propertysecond-order optimality conditions

MSC classification

Primary: 26B25: Convexity, generalizations
Secondary: 49J40: Variational methods including variational inequalities 49J52: Nonsmooth analysis 49J53: Set-valued and variational analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 2 , October 2020 , pp. 217 - 229
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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References

Aussel, D., Corvellec, J. N. and Lassonde, M., ‘Subdifferential characterization of quasiconvexity and convexity’, J. Convex Anal. 1(2) (1994), 195–202.Google Scholar
Aussel, D., Daniilidis, A. and Thibault, L., ‘Subsmooth sets: functional characterizations and related concepts’, Trans. Amer. Math. Soc. 357(4) (2005), 1275–1301.Google Scholar
Borwein, J. M. and Zhu, Q. J., Techniques of Variational Analysis (Springer, New York, 2005).Google Scholar
Burachik, R. S. and Iusem, A. N., Set-valued Mappings and Enlargements of Monotone Operators, Optimization and its Applications, 8 (Springer, New York, 2008).Google Scholar
Chieu, N. H., Chuong, T. D., Yao, J. C. and Yen, N. D., ‘Characterizing convexity of a function by its Fréchet and limiting second-order subdifferentials’, Set-Valued Var. Anal. 19(1) (2011), 75–96.Google Scholar
Chieu, N. H. and Huy, N. Q., ‘Second-order subdifferentials and convexity of real-valued functions’, Nonlinear Anal. 74(1) (2011), 154–160.Google Scholar
Chieu, N. H., Lee, G. M., Mordukhovich, B. S. and Nghia, T. T. A., ‘Coderivative characterizations of maximal monotonicity for set-valued mappings’, J. Convex Anal. 23(2) (2016), 461–480.Google Scholar
Crouzeix, J. P. and Ferland, J. A., ‘Criteria for differentiable generalized monotone maps’, Math. Program. 75(3) (1996), 399–406.10.1007/BF02592191Google Scholar
Daniilidis, A. and Georgiev, P., ‘Approximate convexity and submonotonicity’, J. Math. Anal. Appl. 291(1) (2004), 292–301.Google Scholar
Drusvyatskiy, D. and Lewis, A. S., ‘Tilt stability, uniform quadratic growth and strong metric regularity of the subdifferential’, SIAM J. Optim. 23(1) (2013), 256–267.10.1137/120876551Google Scholar
Gfrerer, H. and Mordukhovich, B. S., ‘Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions’, SIAM J. Optim. 25(4) (2015), 2081–2119.Google Scholar
Levy, A., Poliquin, R. A. and Rockafellar, R. T., ‘Stability of locally optimal solutions’, SIAM J. Optim. 10(2) (2000), 508–604.Google Scholar
Luc, D. T. and Schaible, S., ‘Generalized monotone monsmooth maps’, J. Convex Anal. 3(2) (1996), 195–205.Google Scholar
Mordukhovich, B. S., ‘Sensitivity analysis in nonsmooth optimization’, in: Theoretical Aspects of Industrial Design, SIAM Proceedings in Applied Mathematics, 58 (eds. Field, D. A. and Komkov, V.) (SIAM Publications, Philadelphia, PA, 1992), 32–42.Google Scholar
Mordukhovich, B. S., Variational Analysis and Generalized Differential I, II (Springer, New York, 2006).Google Scholar
Mordukhovich, B. S., Variational Analysis and Applications (Springer, New York, 2018).Google Scholar
Mordukhovich, B. S. and Nghia, T. T. A., ‘Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces’, Nonlinear Anal. 86 (2013), 159–180.Google Scholar
Ngai, H. V., Luc, D. T. and Thera, M., ‘Approximate convex functions’, J. Nonlinear Convex Anal. 1(2) (2000), 155–176.Google Scholar
Poliquin, R. A. and Rockafellar, R. T., ‘Tilt stability of a local minimum’, SIAM J. Optim. 8(2) (1998), 287–299.Google Scholar
Poliquin, R., Rockafellar, R. T. and Thibault, L., ‘Local differentiability of distance functions’, Trans. Amer. Math. Soc. 352(11) (2000), 5231–5249.Google Scholar
Rockafellar, R. T. and Wets, R. J., Variational Analysis (Springer, New York, 1998).Google Scholar
Schirotzek, W., Nonsmooth Analysis (Springer, Berlin, 2007).Google Scholar
Spingarn, J. E., ‘Submonotone subdifferentials of Lipschitz functions’, Trans. Amer. Math. Soc. 264(1) (1981), 77–89.Google Scholar