Abstract
Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionx→‖∇f(x)‖−1∇f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)⩽f(x)} can be expressed as the convex hull of the limits of type {N(x n)}, where {x n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.
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References
Borde, J.,Quelques Aspects Théoriques et Algorithmiques en Quasiconvexité, Doctoral Thesis, University of Clermont, Aubière, France, 1985.
Crouzeix, J. P.,A Review of Continuity and Differentiability Properties of Quasiconvex Functions on R n, Convex Analysis and Applications, Edited by J. P. Aubin and R. Vinter, Pitman, New York, New York, pp. 18–34, 1982.
Chabrillac, Y., andCrouzeix, J. P.,Continuity and Differentiability Properties of Monotone Real Variables, Mathematical Programming Study, Vol. 30, pp. 1–16, 1987.
Crouzeix, J. P., andFerland, J. A.,Criteria for Quasiconvexity and Pseudoconvexity: Relationships and Comparisons, Mathematical Programming, Vol. 23, pp. 193–205, 1982.
Auslender, A.,Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.
Crouzeix, J. P.,Some Differentiability Properties of Quasiconvex Functions on R n, Optimization and Optimal Control, Edited by A. Auslender, W. Oettli, and J. Stoer, Springer-Verlag, Berlin, Germany, pp. 9–20, 1981.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Diewert, W. E.,Alternative Characterization of Six Kinds of Quasiconcavity in the Nondifferentiable Case with Applications to Nonsmooth Programming, Generalized Concavity in Optimization Theory and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, pp. 51–93, 1981
Diewert, W. E.,Duality Approaches to Microeconomic Theory, Handbook of Mathematical Economics, Vol. 2, Edited by K. J. Arrow and M. D. Intriligator, North-Holland, Amsterdam, Holland, pp. 539–599, 1982.
Crouzeix, J. P.,Duality between Direct and Indirect Utility Functions, Journal of Mathematical Economics, Vol. 12, pp. 149–165, 1983.
Martinez-Legaz, J. E.,Duality between Direct and Indirect Utility Functions under Minimal Hypotheses, Journal of Mathematical Economics (to appear).
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Communicated by O. L. Mangasarian
This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.
The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.
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Borde, J., Crouzeix, J.P. Continuity properties of the normal cone to the level sets of a quasiconvex function. J Optim Theory Appl 66, 415–429 (1990). https://doi.org/10.1007/BF00940929
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DOI: https://doi.org/10.1007/BF00940929