Kuhn-Tucker type optimality conditions for some class of nonsmooth programming problems

• Autorzy: Śmietański M. J.
• Języki publikacji: EN

Abstrakty

EN

We consider two nonlinear programming problems with nonsmooth functions. The necessary and sufficient first order optimality conditions use the Dini and Clarke derivatives. However, the obtained Kuhn-Tucker conditions have a rather classical form. The sufficient conditions alone are obtained thanks to some properties of generalized convexity and generalized linearity of functions. The necessary and sufficient optimality conditions are given in the Lagrange form.

Słowa kluczowe

EN

nonsmooth programming; Dini derivative; Clarke derivative; generalized convexity; generalized linearity; necessary and sufficient optimality conditions

PL

pochodna Diniego; pochodna Clarka; wypukłość uogólniona; liniowość uogólniona; warunki konieczne i wystarczające optymalności

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2003
• Tom: Vol. 32, no 2
• Strony: 361--376
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Śmietański M. J.

• Faculty of Mathematics of Łódź University, Banacha 22, 90-238 Łódź, Poland

Bibliografia

• BAZARAA, M.S., SHERALI, H.D. and SHETTY, C.M. (1993) Nonlinear programming. Theory and Algorithms. New York, Second Edition, John Wiley & Sons, Inc.
• BAZARAA, M.S. and SHETTY, C.M. (1976) Foundations of Optimization. Berlin, Springer Verlag.
• CHEW, K.L. and CHOO, E.V. (1984) Pseudolinearity and efficiency. Mathematical Programming, 28, 226-239.
• CLARKE, F .H. (1975) Generalized gradients and applications. Transactions of the American Mathemat·ical Soc·iety, 205, 247-262.
• CLARKE, F.H. (1976) A new approach to Lagrange multipliers. Math. Ope1'. Res., 1, 165-174.
• CRAVEN, B.D. and WANG, S. (preprint) Second or·der optimality conditions ·in multiobjective programming. Australia, University of Melbourne.
• DIEWERT, W.E. (1981) Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to non-smooth programming. In: Generalized Concavity in Opt-imizat·ion and Economics, S. Schaible and W.T. Ziemba (eds.), New York, Academic Press, 51-94.
• ELLAIA, R. and HASSOUNI, A. (1991) Characterizations of nonsmooth functions through their generalized gradients. Optimization, 22 (3), 401-416.
• ELSTER, K.-H. and THIERFELDER, J. (1989) On cone approximations and generalized directional derivatives. In: Nonsmooth Optimization and Related Topics, F.H. Clarke, V.F. Dem'yanov, F. Giannessi (eds.), Plenum Press, New York, 133-154.
• GIANESSI, F. (1989) Semidifferentiable functions and necessary optimality conditions. Journal of Optimization Theory and Applications, 60, 2, 191-241.
• GIORGI G. and GUERRAGGIO, A. (1994) First order generalized optimality conditions for programming problems with a set constraints. In: Generalized Convexity Proceedings, Pees, Hungary, Lect. Notes in Econ. and Math. Syst., 405, Springer-Verlag, 171-185.
• GIORGI, G. and KOMLOSI, S. (1995) Dini derivatives in optimization- part III. Rivista di matematica per la scienze economiche e sociali, 18°, Fasciolo 1°, 47-63.
• GIORGI, G. and MITITELU, S. (1983) Extremum conditions in quasi-convex programming. Dipartimento di Richerca Operativa e Scienze Statistiche,
• GLOVER, B.M. (1984) Generalized convexity in nondifferentiable programming. Bulletin of The Austr-al. Math. Soc ., 30, 193-218.
• KOMLOSI, S. (1993) Quasiconvex first- order approximations and Kuhn-Tucker type optimality conditions. European Joumal of Oper·at·ional ReseaTch, 65, 248-276.
• KOMLOSI, S. (1993) First and second characterizations of pseudolinear functions. Eumpean Joumal of Operational ReseaTch, 67, 278-286.
• KOMLOSI, S. (1994) Generalized monot onicity in non-smooth analysis, in: Lecture Notes in Econ. and Math. Systems , Hungary, Springer-Verlag, 263-275.
• KOMLOSI, S. (1995) Monotonicity and quasimonotonicity in nonsmooth analysis. In: D.-Z. Du, L. Qi, R.S. Womersley, eds., Recent Advances in Nonsmooth Optirn·ization. World Scientific Publishing Co. P te Ltd ., 193- 214.
• KORTANEK, K.O. and EVANS, J .P. (1 96 7) Pseudoconcave Programming and Lagrange regularity. Oper-ations Research, 15, 882-892.
• MANGASARIAN, O.L. (1969) Nonlinear prograrnming. New York, McGraw-Hill Book Co.
• McCORMICK , G. P . (1967) Second-order conditions for constrained minima. SIAM J. Appl. Math., 15, 3, 641-65 2.
• MITITELU, S. (1987) Optimality, mini max and duality in nonlinear programming. Optimization 18, 501-506.
• MITITELU, S. (1994) A survey on optim ality and duality in nonsmooth programming. In: Generalized Convexity, S. Koml6si, T. Rapcsak and S. Scheible (eds.), Springer, Heidelberg, 211-228.
• PENOT, J.P. (1998) Are generalized derivatives useful for generalized convex functions. In: J.-P. Crouzeix, M. Volle, J.-E. Martine,-;-Legaz, eds., Generalized Convexity, Generalized Monotonicity. Kluwer Academic Publishers, 3-39.
• ZANGWILL, W.I. (1969) NonlineaT Pmgmmm ·ing; a Unified Approach. New York, Englewood Cliffs, Prentice-Hall.