Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T22:20:47.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized differentiability for a class of nondifferentiable operators with applications to nonsmooth optimization

Published online by Cambridge University Press:  09 April 2009

Thomas W. Reiland
Thomas W. Reiland
Affiliation:
Department of Statistics and Graduate, Program in Operations Research, North Carolina State University, Box 8203, Raleigh, North Carolina 27695-8203, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A theory of generalized gradients is presented for a class of Lipschitz vector-valued mappings from a Banach space to a locally convex order complete vector lattice. Necessary optimality conditions are obtained for nonconvex programming problems on Banach spaces with vector- valued operator constraints and/or an arbitrary set constraint. Sufficient optimality conditions are also obtained under mild convexity assumptions.

MSC classification

Secondary: 90C48: Programming in abstract spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 1 , August 1989 , pp. 114 - 132
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Borwein, J. M., ‘Continuity and differentiability properties of convex operators’, Proc. London Math. Soc. (3) 44 (1982), 420–444.CrossRefGoogle Scholar
[2]Bourbaki, N., Varietes differentielles et analytiques, elements de mathematiques XXXIII (Hermann, Paris, 1967).Google Scholar
[3]Clarke, F. H., Necesary conditions for nonsmooth problems in optimal control and the calculus of variations, (Ph.D. Thesis, University of Washington, Seattle, 1973).Google Scholar
[4]Clarke, F. H., ‘Generalized gradients and applications’, Trans. Amer. Math. Soc. 205 (1975), 247–262.CrossRefGoogle Scholar
[5]Clarke, F. H., ‘A new approach to Lagrange multipliers’, Math. Oper. Res. 2 (1976), 165–174.CrossRefGoogle Scholar
[6]Clarke, F. H., ‘Generalized gradients of Lipschitz functionals’, Adv. in Math. 40 (1981), 52–66.CrossRefGoogle Scholar
[7]Clarke, F. H., Optimization and nonsmooth analysis, (Wiley and Sons, New York, 1983).Google Scholar
[8]Chancy, R. W., ‘On sufficient conditions in nonsmooth optimization’, Math. Oper. Res. 7 (1982), 463–475.CrossRefGoogle Scholar
[9]Dunford, N. and Schwartz, J. T., Linear operators, Part I: General theory, (Wiley and Sons, New York, 1964).Google Scholar
[10]Guignard, M., ‘Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space’, SIAM J. Control Optim. 7 (1969), 232–241.CrossRefGoogle Scholar
[11]Halkin, H., ‘Interior mapping theorem with set-valued derivatives’, J. Analyse Math. 30 (1976), 200–207.CrossRefGoogle Scholar
[12]Hiriart-Urruty, J. B., ‘On optimality conditions in nondifferentiable programming’, Math. Programming 14 (1978), 73–86.CrossRefGoogle Scholar
[13]Hiriart-Urruty, J. B., ‘Refinements of necessary optimality conditions in nondifferentiable programming I’, Appl. Math. Optim. 5 (1979), 63–82.CrossRefGoogle Scholar
[14]Hiriart-Urruty, J. B., ‘Tangent cones, generalized gradients and mathematical programming in Banach spaces,’ Math. Oper. Res. 4 (1979), 79–97.CrossRefGoogle Scholar
[15]Hiriart-Urruty, J. B., ‘New concepts in nondifferentiable programming,’ Bull. Soc. Math. France 50 (1979), 57–85.Google Scholar
[16]Hiriart-Urruty, J. B., ‘Approximating a second-order directional derivative for nonsmooth convex functions,’ SIAM J. Control Optim. 20 (1982), 783–807.CrossRefGoogle Scholar
[17]Hiriart-Urruty, J. B., ‘Limiting behaviour of the approximate first-order and second-order directional derivatives for a convex function’, Nonlinar Anal. 6 (1982), 1309–1326.CrossRefGoogle Scholar
[18]Hiriart-Urruty, J. B., Strodiot, J. J. and Nguyen, V. H., ‘Generalized hessian matrix and second-order optimality conditions for problems with C1,1 data’, Appl. Math. Optim. 11 (1984), 43–56.CrossRefGoogle Scholar
[19]Ioffe, A. D., ‘Necessary and sufficient conditions for a local minimum. 1: A reduction theorem and first-order conditions,’ SIAM J. Control Optim. 17 (1979), 245–250.CrossRefGoogle Scholar
[20]Ioffe, A. D., ‘Nonsmooth analysis: differential calculus of nondifferentiable mappings,’ Trans. Amer. Math. Soc. 266 (1981), 1–56.CrossRefGoogle Scholar
[21]Ioffe, A. D., ‘Nonsmooth analysis and the theory of fans,’ Convex analysis and optimization edited by Aubin, J. P. and Vinter, R. B., (Pitman, Marshfield, Mass., 1982), pp. 93–117.Google Scholar
[22]Ioffe, A. D., ‘Necessary conditions in nonsmooth optimization,’ Math. Oper. Res. 9 (1984), 159–189.CrossRefGoogle Scholar
[23]Ioffe, A. D. and Levin, L., ‘Subdifferentials of convex functions’, Trans. Moscow Math. Soc. 26 (1972), 1–72.Google Scholar
[24]Kusraev, A. G., ‘On necessary conditions for an extremum of nonsmooth vector-valued mappings,’ Soviet Math. Dokl. 19 (1978), 1057–1060.Google Scholar
[25]Kutateladze, S. S., ‘Support sets of sublinear operators’, Dokl. Akad. Nauk SSSR 230 (1976), 1029–1032.Google Scholar
[26]McLinden, L., ‘Dual operations on saddle functions,’ Trans. Amer. Math. Soc. 179 (1973), 363–381.CrossRefGoogle Scholar
[27]Moreau, J. J., Fonctionelles convexes, (Lecture notes, Seminaire Equations aux Derivees Partielles, College de France, 1966).Google Scholar
[28]Papageorgiou, N. S., ‘Nonsmooth analysis on partially ordered vector spaces: part I: Convex case,’ Paciflc J. Math. 107 (1983), 403–458.CrossRefGoogle Scholar
[29]Papageorgiou, N. S., Nonsmooth analysis on partially ordered vector spaces: Part 2: Non- convex case, Clarke's theory,’ Pacific J. Math. 109 (1983), 463–495.CrossRefGoogle Scholar
[30]Penot, J. P., ‘Calcul sous-differentiels et optimisation,’ I Funct. Anal. 27 (1978), 248–276.CrossRefGoogle Scholar
[31]Peressini, A. L., Ordered topological vector spaces (Harper and Row, New York, 1967).Google Scholar
[32]Reiland, T. W. and Chou, J. H., Saddlepoint problems and duality in nondifferential programming, (Report #205, North Carolina State University Graduate Program in Operations Research, 1984).Google Scholar
[33]Robinson, S. M., ‘Regularity and stability theorems for convex multivalued functions,’ Math. Oper. Res. 1 (1976), 130–142.CrossRefGoogle Scholar
[34]Rockafellar, R. T., Convex analysis (Princeton Univ. Press, Princeton, N.J., 1970).CrossRefGoogle Scholar
[35]Rockafeller, R. T., ‘Conjugate convex functions in optimal control and the calculus of variations’, J. Math. Anal. Appl 32 (1970), 174–222.CrossRefGoogle Scholar
[36]Rockafeller, R. T., ‘Existence and duality theorems for convex problems of Bolza’, Trans. Amer. Math. Soc. 159 (1971), 1–40.CrossRefGoogle Scholar
[37]Rockafeller, R. T., ‘Integrals which are convex functionals, II’, Pacific J. Math. 39 (1971), 439–469.CrossRefGoogle Scholar
[38]Rockafeller, R. T., ‘Conjugate duality and optimization (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974).CrossRefGoogle Scholar
[39]Rockafeller, R. T., ‘Clarke's tangent cones and the boundaries of closed sets in Rn’, Nonlinear Anal 3 (1979), 145–154.CrossRefGoogle Scholar
[40]Rockafeller, R. T., ‘Directionally Lipschitzian functions and subdifferential calculus,’ Proc. London Math. Soc. 39 (1979), 331–355.CrossRefGoogle Scholar
[41]Rockafeller, R. T., ‘Generalized directional derivatives and subgradients of nonconvex functions,’ Canad. J. Math. 22 (1980), 257–280.CrossRefGoogle Scholar
[42]Rockafeller, R. T., The theory of subgradients and its applications to problems of optimization. Convex and nonconvex functions (Heldermann-Verlag, Berlin, 1981).Google Scholar
[43]Rockafeller, R. T., ‘Marginal values and second-order necessary conditions for optimality,’ Math. Programming 26 (1983), 245–286.CrossRefGoogle Scholar
[44]Rubinov, A. M., ‘Sublinear operators and their applications,’ Russian Math. Surveys 32 (1977), 115–175.CrossRefGoogle Scholar
[45]Schaefer, H. H., Topological vector spaces (Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
[46]Thibault, L., ‘Fonctions compactement Lipschitziennes et programmation mathematique,’ C. R. Acad. Sci. Paris Ser. A 287 (1978), 213–216.Google Scholar
[47]Thibault, L., Subditferentials of compactly Lipschitzian vector-valued functions,’ Ann. Mat. Pura Appl. 125 (1980), 157–192.CrossRefGoogle Scholar
[48]Thibault, L., ‘Subdifferentials of nonconvex vector-valued functions,’ J. Math. Anal. Appl. 86 (1982), 319–344.CrossRefGoogle Scholar
[49]Thibault, L., ‘On generalized differentials and subdifferentials of Lipschitz vector-valued functions,’ Nonlinear Anal 6 (1982), 1037–1053.CrossRefGoogle Scholar
[50]Thibault, L., ‘Tangent cones and quasi-interiorly tangent cones to multifunctions,’ Trans. Amer. math. Soc. 277 (1983), 601–62.CrossRefGoogle Scholar
[51]Valadier, M., ‘Sous-differentiabilite de fonctions convexes a valeurs dans un espace vectoriel ordonne,’ Math. Scand. 30 (1972), 65–74.CrossRefGoogle Scholar
[52]Varaiya, P. P., ‘Nonlinear programming in Banach space,’ SIAM J. Appl. Math. 15 (1967), 284–293.CrossRefGoogle Scholar
[53]Warga, J., Derivative containers, inverse functions and controllability, calculus of variations and control theory, (Russell, D. L., ed., Academic Press, New York, 1976).Google Scholar
[54]Zowe, J., ‘Subdifferentiability of convex functions with values in an ordered vector space,’ Math. Scand. 34 (1974), 69–83.CrossRefGoogle Scholar
[55]Zowe, J., ‘A duality theorem for a convex programming problem in order complete vector lattices,’ J. Math. Anal. Appl 50 (1975), 273–287.CrossRefGoogle Scholar