We observe that a quasiconvex function which is evenly quasiconvex at a point is not necessarily Greenberg–Pierskalla (briefly, G-P) subdifferentiable at that point, but we prove that a quasiconvex function which is upper semicontinuous on the segments of its effective domain is G-P subdifferentiable on the relative interior of this effective domain. We give an application to surrogate duality in quasiconvex programming.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aussel, D., Daniilidis, A.: Normal characterization of the main classes of quasiconvex functions. Set-Valued Anal. 8, 219–236 (2000)
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)
Crouzeix, J.P.: Conjugacy in quasiconvex analysis. In: Auslender, A (ed.) Convex Analysis and Its Applications. Lecture Notes in Economics and Mathematical Systems, vol. 144, pp 66–99. Springer, Berlin (1977)
Crouzeix, J.P.: Contributions a l’Étude des Fonctions Quasiconvexes. Ph.D. thesis Université de Clermont-Ferrand, vol. II. Aubière cedex, France (1977)
Crouzeix, J.-P.: Continuity and differentiability properties of quasiconvex functions. In: Schaible, S., Ziemba, W. T. (eds.) Generalized Concavity in Optimization and Economics, pp 109–130. Academic Press, New York (1981)
Daniilidis, A., Hadjisavvas, N., Martínez-Legaz, J.E.: An appropriate subdifferential for quasiconvex functions. SIAM J. Optim. 12, 407–420 (2001)
dos Santos Gromicho, J.A.: Quasiconvex Optimization and Location Theory. Applied Optimization, vol. 9. Kluwer Academic Publishers, Dordrecht (1998)
Goberna, M.A., Jornet, V., Rodríguez, M.M.L.: On linear systems containing strict inequalities. Linear Algebra Appl. 360, 151–171 (2003)
Greenberg, H.J., Pierskalla, W.P.: Surrogate mathematical programming. Oper. Res. 18, 924–939 (1970)
Greenberg, H.J., Pierskalla, W.P.: Quasi-conjugate functions and surrogate duality. Cah. Cent. Études Rech. Opér. 15, 437–448 (1973)
Kiwiel, K.C.: Convergence and efficiency of subgradient methods for quasiconvex minimization. Math. Program. Ser. A 90, 1–25 (2001)
Luenberger, D.G.: Quasi-convex programming. SIAM J. Appl. Math. 16, 1090–1095 (1968)
Martínez-Legaz, J.-E.: A generalized concept of conjugation. In: Hiriart-Urruty, J.-B., Oettli, W., Stoer, J (eds.) Optimization: Theory and Algorithms. Lecture Notes in Pure and Applied Mathematics, vol. 86, pp 45–59. Dekker, New York (1983)
Martínez-Legaz, J.-E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)
Martínez-Legaz, J.E.: Weak lower subdifferentials and applications. Optimization 21, 321–341 (1990)
Martínez-Legaz, J.E.: Generalized convex duality and its economic applications. In: Hadjisavvas, N., Komlósi. S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and Its Applications, vol. 76, pp 237–292. Springer, New York (2005)
Martínez-Legaz, J.E., Romano-Rodríguez, S.: α-lower subdifferentiable functions. SIAM J. Optim. 3, 800–825 (1993)
Martínez-Legaz, J.E., Sach, P.H.: A new subdifferential in quasiconvex analysis. J. Convex Anal. 6, 1–12 (1999)
Penot, J.-P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990)
Penot, J.-P., Volle, M.: Surrogate programming and multipliers in quasi-convex programming. SIAM J. Control Optim. 42, 1994–2003 (2004)
Plastria, F.: Lower subdifferentiable functions and their minimization by cutting planes. J. Optim. Theory Appl. 46, 37–53 (1985)
Ramík, J., Vlach, M.: Generalized Concavity in Fuzzy Optimization and Decision Analysis. International Series in Operations Research & Management Science, vol. 41. Kluwer Academic Publishers, Boston (2002)
Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)
Schaible, S., Ziemba, W.T. (eds.): Generalized Concavity in Optimization and Economics, vol. 1980. Academic Press, New York (1981)
Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for non-essentially quasiconvex programming. Optim. Lett. 11, 1699–1712 (2017)
Volle, M.: Conjugaison par tranches. Ann. Mat. Pura Appl. 139, 279–311 (1985)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing, River Edge (2002)
J. E. Martínez-Legaz acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through Grant PGC2018-097960-B-C21 and the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563). He is affiliated with MOVE (Markets, Organizations and Votes in Economics).
We are grateful to two anonymous referees for their helpful remarks.
Dedicated to Professor Marco A. López on the occasion of his 70th birthday.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Volle, M., Martínez-Legaz, J.E. Some Remarks on Greenberg–Pierskalla Subdifferentiability of Quasiconvex Functions. Vietnam J. Math. 48, 391–406 (2020). https://doi.org/10.1007/s10013-020-00391-6
- Quasiconvex functions
- Greenberg–Pierskalla subdifferential
- Quasiconvex programming
- Surrogate duality
Mathematics Subject Classification (2010)