Abstract
Broadly speaking, a generalized convex function is one which has some property of convex functions that is essential in a particular application. Two such properties are convexity of lower level sets (in the case of quasiconvex functions) and convexity of the ordinary directional derivative as a function of direction (in the case of Pshenichnyi’s quasidifferentiable functions). In recent years, several directional derivatives have been defined that, remarkably, are always convex as a function of direction.
This means that all functions are “generalized convex” in the sense that they have certain convex directional derivatives. As a result, it has become worthwhile to develop generalizations of the Fritz John and Kuhn-Tucker optimality conditions in terms of the subgradients of convex directional derivatives. In this paper, we derive some general versions of these conditions for an inequality-constrained, nondifferentiable, nonconvex mathematical program.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Borvein J.M., Fractional programming without differentiability. Mathematical Programming 11 (1976) 283–290.
Borwein J.M., Stability and regular points of inequality systems. Journal of Optimization Theory and Applications 48 (1986) 9–52.
Borwein J.M.. Fitzpatrick S.P., and Giles J.R., The differentiability of real functions on normed linear Space using generalized subgradients. Journal of Mathematical Analysis and Applications 128 (1987) 512–534.
Borwein J.M. and Strojwas H.M., Directionally Lipschitzian mapping on Baire spaces. Canadiana Journal of Mathematics 36 (1984) 95–130.
Caligaris O. and Oliva P., Necessary conditions for local Pareto minima. Bollettino della Unione Matematica Italiana 5-B (1986) 721 -750.
Clarke FJBL, A new approach to Lagrange multipliers. Mathematics of Operations Research 1 (1976) 165–174.
Clarke F.H., Optimization and nonsmooth analysis. John Wiley and Sons, New York, 1983.
Ekeland I. and Temam R., Convex analysis and variational problems. North- Holland, Amsterdam, 1976.
Hiriart-Urruty J.-B., On optimality conditions in nondifferentiable programming. Mathematical Programming 14 (1978) 73–86.
Hiriart-Urruty J.-B., New concepts in nondifferentiable programming. Bull. Soc. Math. France, Memoire 60 (1979) 57–85.
Ioffe A.D., Neressary condition for a local minimum. A reduction theorem and first-order conditions. STAM J.Cont. Opt. 17 (1979) 245–250.
Jeyakumar V., On optimality conditions in nonsmooth inequality-constrained minimization. Numerical Functional Analysis and Optimization 9 (1987) 535–546.
Merkovsky R.R. and Vard D.E., Upper D.S.L. approximates and nonsmooth optimization. preprint, 1988.
Michel P. and Penot J.-P., Calcul sous-differentiel pour des fonctions lipschitziennes et non lipschitziennes. C.R. Acad. Sci. Paris 298 (1984) 269–272.
Penot J.-P., Variations on the theme of nonsmooth analysis: another subdifferential. in: Demyanov V.F. and Pallaschke D., eds., Nondifferentiable optimization: motivations and applications, Springer-Verlag, Berlin, 1985, pp. 41–54.
Penot J.-P. and Terpolilli P., Cone tangents et singularites. C.R. Acad. Sci. Paris 296 (1983)721–724.
Pschenichnyi B.N., Necessary Conditions for an extremum. Marcel Dekker, New York, 1971.
Pschenichnyi B.N. and Khachatryan R.A., Constraints of equality type in nonsmooth optimization problems. Soviet Mathematics Doklady 26, (1982) 659–662.
Rockafellar R.T., Directionallv Lipschitzian functions and subdifferential calculus. Proceedings of the London Mathematical Society 39 (1979) 331–355.
Rockafellar R.T., Generalized directional derivative and subgradients of nonconve functions. Canadian Journal of Mathematics 32 (1980) 257–280.
Studniarski M., Mean value theorems and sufficient optimality conditions for nonsmooth functions. Journal of Mathematical Analysis and Applications 111(1985)313–326.
Ward D E., Convex subcones of the contingent cone in nonsmooth calculus and optimization. Transactions of the Américain Mathematical Society 302 (1987) 661–682.
Ward D.E., Directional derivative calculus and ootimalitv conditions in nonsmooth mathematical programming,to appear in Journal of Information and Optimization Sciences.
Ward D.B. and Borvein J.M., Nonsmooth callus in finite dimensions. SIAM J. Cont. Opt. 25 (1987) 1312–1340.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ward, D. (1990). Convex Directional Derivatives in Optimization. In: Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P., Schaible, S. (eds) Generalized Convexity and Fractional Programming with Economic Applications. Lecture Notes in Economics and Mathematical Systems, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46709-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-46709-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52673-5
Online ISBN: 978-3-642-46709-7
eBook Packages: Springer Book Archive