We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim. 41 (2008) 1-13] and [Ioffe, Math. Surveys 55 (2000) 501-558; Control Cybern. 32 (2003) 543-554]) constitute the starting point of the present work.
Mots-clés : convex semi-infinite programming, modulus of metric regularity, d.c. functions
@article{COCV_2009__15_4_763_0,
author = {C\'anovas, Mar{\'\i}a J. and Hantoute, Abderrahim and L\'opez, Marco A. and Parra, Juan},
title = {Lipschitz modulus in convex semi-infinite optimization via d.c. functions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {763--781},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
year = {2009},
doi = {10.1051/cocv:2008052},
mrnumber = {2567244},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008052/}
}
TY - JOUR AU - Cánovas, María J. AU - Hantoute, Abderrahim AU - López, Marco A. AU - Parra, Juan TI - Lipschitz modulus in convex semi-infinite optimization via d.c. functions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 763 EP - 781 VL - 15 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008052/ DO - 10.1051/cocv:2008052 LA - en ID - COCV_2009__15_4_763_0 ER -
%0 Journal Article %A Cánovas, María J. %A Hantoute, Abderrahim %A López, Marco A. %A Parra, Juan %T Lipschitz modulus in convex semi-infinite optimization via d.c. functions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 763-781 %V 15 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008052/ %R 10.1051/cocv:2008052 %G en %F COCV_2009__15_4_763_0
Cánovas, María J.; Hantoute, Abderrahim; López, Marco A.; Parra, Juan. Lipschitz modulus in convex semi-infinite optimization via d.c. functions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 763-781. doi : 10.1051/cocv:2008052. http://www.numdam.org/articles/10.1051/cocv:2008052/
[1] and , Weak sharp minima in mathematical programming. SIAM J. Contr. Opt. 31 (1993) 1340-1359. | MR | Zbl
[2] , and , On the Lipschitz modulus of the argmin mapping in linear semi-infinite optimization. Set-Valued Anal. (2007) Online First. | MR | Zbl
[3] , , and , Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. 18 (2007) 717-732. | MR
[4] , , and , Lipschitz behavior of convex semi-infinite optimization problems: A variational approach. J. Global Optim. 41 (2008) 1-13. | MR
[5] , , and , Stability of indices in the KKT conditions and metric regularity in convex semi-infinite optimization. J. Optim. Theory Appl. (2008) Online First. | MR
[6] , , and , Lipschitz modulus of the optimal set mapping in convex semi-infinite optimization via minimal subproblems. Pacific J. Optim. (to appear). | Zbl
[7] , and , Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Acad. Nat. Lincei, Rend, Cl. Sci. Fiz. Mat. Natur. 68 (1980) 180-187. | MR | Zbl
[8] and , Quasidifferentiable functionals. Dokl. Akad. Nauk SSSR 250 (1980) 21-25 (in Russian). | MR | Zbl
[9] and , Constructive nonsmooth analysis, Approximation & Optimization 7. Peter Lang, Frankfurt am Main (1995). | MR | Zbl
[10] and , Nonlinear programming. Wiley, New York (1968). | MR | Zbl
[11] and , Linear Semi-Infinite Optimization. John Wiley & Sons, Chichester, UK (1998). | MR | Zbl
[12] and , Convex analysis and minimization algorithms, I. Fundamentals, Grundlehren der Mathematischen Wissenschaften 305. Springer-Verlag, Berlin (1993). | MR | Zbl
[13] , Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55 (2000) 103-162; English translation in Math. Surveys 55 (2000) 501-558. | MR | Zbl
[14] , On rubustness of the regularity property of maps. Control Cybern. 32 (2003) 543-554. | Zbl
[15] and , Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer Academic Publ., Dordrecht (2002). | MR | Zbl
[16] and , Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16 (2005) 96-119. | MR | Zbl
[17] and , A note of Lipschitz constants for solutions of linear inequalities and equations. Linear Algebra Appl. 244 (1996) 365-374. | MR | Zbl
[18] , Approximation et Optimisation. Hermann, Paris (1972). | MR | Zbl
[19] , The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program. Linear Algebra Appl. 187 (1993) 15-40. | MR | Zbl
[20] , Variational Analysis and Generalized Differentiation I. Springer-Verlag, Berlin (2006). | MR | Zbl
[21] , Unicity in semi-infinite optimization, in Parametric Optimization and Approximation, B. Brosowski, F. Deutsch Eds., Birkhäuser, Basel (1984) 231-247. | MR | Zbl
[22] , Bounds for error in the solution set of a perturbed linear program. Linear Algebra Appl. 6 (1973) 69-81. | MR | Zbl
[23] , Convex Analysis. Princeton University Press, Princeton, USA (1970). | MR | Zbl
[24] and , Variational Analysis. Springer-Verlag, Berlin (1997). | MR | Zbl
[25] and , Weak sharp minima: Characterizations and sufficient conditions. SIAM J. Contr. Opt. 38 (1999) 219-236. | MR | Zbl
[26] , Sous-différentiels d'une borne supérieure et d'une somme continue de fonctions convexes. C. R. Acad. Sci. Paris 268 (1969) 39-42. | MR | Zbl
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