Abstract
Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fréchet subdifferentials in Asplund spaces. The first criterion extends to such spaces a result established by Barron et al. (Discrete Contin Dyn Syst Ser B 17:1693–1706, 2012). The second criterion is totally new even if it is applied to lower semicontinuous functions on finite-dimensional spaces.
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References
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Correa, R., Jofré, A., Thibault, L.: Characterization of lower semicontinuous convex functions. Proc. Am. Math. Soc. 116, 67–72 (1992)
Correa, R., Jofré, A., Thibault, L.: Subdifferential characterization of convexity. In: Du, D., Qi, L., Womersley, R. (eds.) Recent Advances in Nonsmooth Optimization, pp. 18–23. World Scientific, Singapore (1995)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)
Poliquin, R.A.: Subgradient monotonicity and convex functions. Nonlinear Anal. 14, 305–317 (1990)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
Aussel, D., Corvellec, J.-N., Lassonde, M.: Subdifferential characterization of quasiconvexity and convexity. J. Convex Anal. 1, 195–201 (1994)
Aussel, D., Corvellec, J.-N., Lassonde, M.: Mean-value property and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147–4161 (1995)
Aussel, D.: Subdifferential properties of quasiconvex and pseudoconvex functions: unified approach. Optimization 97, 29–45 (1998)
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)
Luc, D.T.: Characterisations of quasiconvex functions. Bull. Aust. Math. Soc. 48, 393–406 (1993)
Penot, J.-P., Quang, P.H.: Generalized convexity of functions and generalized monotonicity of set-valued maps. J. Optim. Theory Appl. 92, 343–356 (1997)
Trang, N.T.Q.: A note on an approximate mean value theorem for Fréchet subgradients. Nonlinear Anal. 75, 380–383 (2012)
Phu, H.X., An, P.T.: Stable generalization of convex functions. Optimization 38, 309–318 (1996)
Barron, E.N., Goebel, R., Jensen, R.R.: Functions which are quasiconvex under linear perturbations. SIAM J. Optim. 22, 1089–1108 (2012)
An, P.T.: Stability of generalized monotone maps with respect to their characterizations. Optimization 55, 289–299 (2006)
Khanh, P.D., Phat, V.T.: Second-order characterizations of \({\cal{C}}^{1}\)-smooth robustly quasiconvex functions. Oper. Res. Lett. 46, 568–572 (2018)
Zagrodny, D.: Approximate mean value theorem for upper subderivatives. Nonlinear Anal. 12, 1413–1428 (1988)
Thibault, L.: A note on the Zagrodny mean value theorem. Optimization 35, 127–130 (1995)
Barron, E.N., Goebel, R., Jensen, R.R.: Quasiconvex functions and nonlinear PDEs. Trans. Am. Math. Soc. 365, 4229–4255 (2013)
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology (NAFOSTED) under Grant 101.01–2017.325, and by Vietnam Institute for Advanced Study in Mathematics (VIASM). Hoa T. Bui is supported by an Australian Government Research Training Program (RTP) Stipend and RTP Fee-Offset Scholarship through Federation University Australia. Pham Duy Khanh is supported by FONDECYT postdoc Grant No. 3180080, and by Basal Program CMM–AFB 170001 from CONICYT–Chile. The authors are grateful to the anonymous referees for their careful reading, encouragement, and valuable suggestions.
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Communicated by Lionel Thibault.
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Bui, H.T., Khanh, P.D. & Tran, T.T.T. Characterizations of Nonsmooth Robustly Quasiconvex Functions. J Optim Theory Appl 180, 775–786 (2019). https://doi.org/10.1007/s10957-018-1421-3
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DOI: https://doi.org/10.1007/s10957-018-1421-3
Keywords
- Quasiconvexity
- Robust quasiconvexity
- Quasimonotone
- Fréchet subdifferential
- Approximate mean value theorem