Address: Markusa st. 22, Vladikavkaz,
362027, RNO-A, Russia
Phone: (8672)50-18-06
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2016.3.5879
Necessary Optimality Conditions in Non-Smooth Problems with Equality Constraints
Khachatryan R. A.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 3.
Abstract: Necessary conditions for extremum in non smooth problems are obtained in this article. The problem under consideration includes both equality and inequality type constrains given by non-smooth functions. The necessary conditions are given in terms of asymptotic subdifferentials. Generalized Lagranges's multiplier rule for non-smooth problems with not local lipschitz constraints is obtained. It is proved also that Peno's and Clark's generalized derivatives are upper convex approximations for local Lipshitz functions.
Bolt`yanskii W. G. Optimlnoe Upravlenie Diskretnimi Sistemam
[Optimal Control of Discret Sistems]. Moskva, 1973, Nauka, 446 p.
Bolt` yanskii V. G. The Method of Tents in theTheory of Extremal
Problems, Russian Mathematical Surveys, 1975, vol. 30, no. 3, pp.
1-54. DOI:10.1070/RM1975v030n03ABEH001411.
Dmitruk A. V., Miluytin A. A. and Osmolovskii N. P. Lyusternik`s
Theorem and the Theory of Extrema, Russian Mathematical Surveys,
vol. 35, no. 6, pp. 11-51. DOI: 10.1070/RM1980v035n06 ABEH001973.
Clarke F. H. A New Approach to Lagrange Multipliers. Math. Oper.
Res. 1., 1976, no 1, pp. 165-174.
Polovinkin E. S. Mnogoznachnij Analiz i Differencialnoe
Wkluchenie [Multivalued Analysis and Diferential Inclusion]. Moskva,
2014, Fizmatlit, 608p.
Pshenichnii B. N.Wypuklyy Analiz i Ekstremal`nyye Zadachi [Convex
analysis and exstremal problems]. Moskva, 1980, Nauka, 320p.
Pshenichnii B. N., Khachatryan R. A. Equality Costraints in
Non-Smooth Optimization Problems, Dokladi Akademii Nauk SSSR,1982,
vol.267, no 3, pp. 553-556 [in Russian].
Pshenichnii B. N., Khachatryan R. A. À Necessary Condition for an
Extremum Nonsmoth Problems, Kibernetika, 1983, no 3, pp. 111-116 [in
Russian] .
Khachatryan R. A. O the Intersection of Tents in Hilbert Space and
the Necessary Optimality Conditions for Nonsmooth Functions,
Izvestia Akademii Nauk ARM SSR, Series Mat., 1988, vol. 23,
no. 3, pp.149-162 [in Russian].
Michel P., Penot J.-P. Calcul Sous Differentiel Pour les Functions
Lipschiziennes et non Lipschiziennes, C.R. Acad. Sc. Paris
Ser.,1984, vol. 291, pp. 269-272.
Ioffe A. D. Lagrange Multiplier rule with small Convex-Valued
Subdifferential for Non-smoth Problems of Mathematical Programming
Involving Equality and Nonfunctional Constraints, Math. Programming,
1993, no. 58, pp. 137-145.
Ivanishi R. On the Intersection of Continous Local Tents, Proc.
Japan Acad. Ser. A, 1993, vol. 69, pp.308-311.
Ekland I. On the Variational Principle, J. Math. Annal. Appl., 1974,
vol. 47, no. 2, pp. 324-353.
