Geodesic
On lower Lipschitz continuity of minimal points
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 20 (2000) no. 2, pp. 245-255.

Voir la notice de l'article provenant de la source Library of Science

In this paper we investigate the lower Lipschitz continuity of minimal points of an arbitrary set A depending upon a parameter u . Our results are formulated with the help of the modulus of minimality. The crucial requirement which allows us to derive sufficient conditions for lower Lipschitz continuity of minimal points is that the modulus of minimality is at least linear. The obtained results can be directly applied to stability analysis of vector optimization problems.
Keywords: minimal points, Lipschitz continuity, vector optimization 
@article{DMDICO_2000_20_2_a5,
     author = {Bednarczuk, Ewa},
     title = {On lower {Lipschitz} continuity of minimal points},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {245--255},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2000},
     zbl = {0977.90047},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2000_20_2_a5/}
}
TY  - JOUR
AU  - Bednarczuk, Ewa
TI  - On lower Lipschitz continuity of minimal points
JO  - Discussiones Mathematicae. Differential Inclusions, Control and Optimization
PY  - 2000
SP  - 245
EP  - 255
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2000_20_2_a5/
LA  - en
ID  - DMDICO_2000_20_2_a5
ER  - 
%0 Journal Article
%A Bednarczuk, Ewa
%T On lower Lipschitz continuity of minimal points
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2000
%P 245-255
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMDICO_2000_20_2_a5/
%G en
%F DMDICO_2000_20_2_a5
Bednarczuk, Ewa. On lower Lipschitz continuity of minimal points. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 20 (2000) no. 2, pp. 245-255. http://geodesic.mathdoc.fr/item/DMDICO_2000_20_2_a5/

[1] T. Amahroq and L. Thibault, On proto-differentiability and strict proto-differentiability of multifunctions of feasible points in perturbed optimization problems, Numerical Functional Analysis and Optimization 16 (1995), 1293-1307.

[2] J.-P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser 1990.

[3] E. Bednarczuk, Berge-type theorems for vector optimization problems, optimization, 32 (1995), 373-384.

[4] E. Bednarczuk, On lower semicontinuity of minimal points, to appear in Nonlinear Analysis, Theory and Applications.

[5] E. Bednarczuk and W. Song, PC points and their application to vector optimization, Pliska Stud. Math. Bulgar. 12 (1998), 1001-1010.

[6] N. Bolintineanu and A. El-Maghri, On the sensitivity of efficient points, Revue Roumaine de Mathematiques Pures et Appliques 42 (1997), 375-382

[7] M.P. Davidson, Lipschitz continuity of Pareto optimal extreme points, Vestnik Mosk. Univer. Ser. XV, Vychisl. Mat. Kiber. 63 (1996), 41-45.

[8] M.P. Davidson, Conditions for stability of a set of extreme points of a polyhedron and their applications, Ross. Akad. Nauk, Vychisl. Tsentr, Moscow 1996.

[9] M.P. Davidson, On the Lipschitz stability of weakly Slater systems of convex inequalities, Vestnik Mosk. Univ., Ser. XV (1998), 24-28.

[10] Deng-Sien, On approximate solutions in convex vector optimization, SIAM Journal on Control and Optimization 35 (1997), 2128-2136.

[11] A. Dontchev and T. Rockafellar, Characterization of Lipschitzian stability, pp. 65-82, Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 1998.

[12] J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Verlag Peter Lang, Frankfurt 1986.

[13] R. Janin and J. Gauvin, Lipschitz dependence of the optimal solutions to elementary convex programs, Proceedings of the 2nd Catalan Days on Applied Mathematics, Presses University, Perpignan 1995.

[14] Wu-Li, Error bounds for piecewise convex quadratic programs and applications, SIAM Journal on Control and Optimization 33 (1995), 1510-1529.

[15] D.T. Luc, Theory of Vector Optimization, Springer Verlag, Berlin 1989.

[16] K. Malanowski, Stability of Solutions to Convex Problems of Optimization, Lecture Notes in Control and Information Sciences 93 Springer Verlag.

[17] E.K. Makarov and N.N. Rachkovski, Unified representation of proper efficiencies by means of dilating cones, JOTA 101 (1999), 141-165.

[18] B. Mordukhovich, Sensitivity analysis for constraints and variational systems by means of set-valued differentiation, Optimization 31 (1994), 13-43.

[19] B. Mordukhovich and Shao Yong Heng, Differential characterisations of convering, metric regularity and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Analysis, Theory, Methods, and Applications 25 (1995), 1401-1424.

[20] D. Pallaschke and S. Rolewicz, Foundation of Mathematical Optimization, Math. Appl. 388, Kluwer, Dordecht 1997.

[21] R.T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Analysis, Theory, Methods and Applications 9 (1985), 867-885.

[22] N. Zheng, Proper efficiency in locally convex topological vector spaces, JOTA 94 (1997), 469-486.

[23] N.D. Yen, Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint, Mathematics of OR 20 (1995), 695-705.

[24] X.Q. Yang, Directional derivatives for set-valued mappings and applications, Mathematical Methods of OR 48 (1998), 273-283.