Abstract
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.
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References
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)
Acknowledgements
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.
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Dedicated to Professor Marco A. López on the occasion of his 70th birthday.
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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
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DOI: https://doi.org/10.1007/s10013-020-00391-6