Abstract
In this paper we characterize those quadratic functions whose restrictions to a convex set are boundedly lower subdifferentiable and, for the case of closed hyperbolic convex sets, those which are lower subdifferentiable but not boundedly lower subdifferentiable.
Once characterized, we will study the applicability of the cutting plane algorithm of Plastria to problems where the objective function is quadratic and boundedly lower subdifferentiable.
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References
M. Avriel, W.E. Diewert, S. Schaible and I. Zang,Generalized Concavity (Plenum, New York, 1988).
J. Bair, “Liens entre le cone d'ouverture interne et l'internat du cone asymptotique d'un convexe,”Bulletin de la Societé Mathématique de Belgique. Série B XXXV (1983) 177–187.
E.J. Balder, “An extension of duality-stability relations to non convex problems,”SIAM Journal on Control and Optimization 15 (1977) 329–343.
M. Boncompte and J.E. Martínez-Legaz, “Fractional programming by lower subdifferentiability techniques,”Journal of Optimization Theory and Applications 68 (1991) 95–116.
J.P. Crouzeix, “Contributions a l'étude des fonctions quasiconvexes,” thesis, Université de Clermont-Ferrand II (Clermont-Ferrand, 1977).
H.P. Greenberg and W.P. Pierskalla, “Quasiconjugate function and surrogate duality,”Cahiers du Centre d'études de Recherche Opérationnelle 15 (1973) 437–448.
D.T. Luc,Theory of Vector Optimization (Springer, Berlin, 1989).
J.E. Martínez-Legaz, “On lower subdifferentiable functions,” in:Trends in Mathematical Optimization. Proceedings, International Conference of Irsee, 1986 (Birkhäuser, Boston, MA, 1988) pp. 197–232.
J.E. Martínez-Legaz, “Quasiconvex duality theory by generalized conjugation methods,”Optimization 19 (1988) 603–652.
J.J. Moreau, “Inf-convolution, sous-additivité, convexité des fonctions numériques,”Journal de Mathématiques Pures et Appliqués 49 (1970) 109–154.
J.P. Penot and M. Volle, “Another duality scheme for quasiconvex problems,” in:Trends in Mathematical Optimization. Proceedings, International Conference of Irsee, 1986 (Birkhäuser, Boston, MA, 1988) pp. 259–275.
F. Plastria, “Lower subdifferentiable functions and their minimization by cutting planes,”Journal of Optimization Theory and Applications 46 (1985) 37–53.
F. Plastria, “The minimization of lower subdifferentiable functions under nonlinear constraints: An all feasible cutting plane algorithm,”Journal of Optimization Theory and Applications 57 (1988) 463–484.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
Y. Sawaragi, H. Nakayama and T. Tanino,Theory of Multiobjective Optimization (Academic Press, New York, 1985).
S. Schaible, “Quasiconvex, pseudoconvex and strictly pseudoconvex quadratic functions,”Journal of Optimization Theory and Applications 35 (1981) 303–338.
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Financial support from the Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058, is gratefully acknowledged.
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Martínez-Legaz, J.E., Romano-Rodríguez, S. Lower subdifferentiability of quadratic functions. Mathematical Programming 60, 93–113 (1993). https://doi.org/10.1007/BF01580603
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DOI: https://doi.org/10.1007/BF01580603