Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T04:26:54.030Z Has data issue: false hasContentIssue false
  • English
  • Français

Lipschitz modulus in convex semi-infinite optimization via d.c.functions

Published online by Cambridge University Press:  19 July 2008

María J. Cánovas ,
Abderrahim Hantoute ,
Marco A. López  and
Juan Parra
María J. Cánovas
Affiliation:
Operations Research Center, Miguel Hernández University of Elche, 03202 Elche, Alicante, Spain. canovas@umh.es; parra@umh.es
Abderrahim Hantoute
Affiliation:
Postdoc researcher at Operations Research Center, Miguel Hernández University of Elche, 03202 Elche, Alicante, Spain. hantoute@ua.es
Marco A. López
Affiliation:
Department of Statistics and Operations Research, University of Alicante, 03071 Alicante, Spain. marco.antonio@ua.es
Juan Parra
Affiliation:
Operations Research Center, Miguel Hernández University of Elche, 03202 Elche, Alicante, Spain. canovas@umh.es; parra@umh.es
Get access

Abstract

We are concerned with the Lipschitz modulus of the optimal set mappingassociated with canonically perturbed convex semi-infinite optimizationproblems. Specifically, the paper provides a lower and an upper bound forthis 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 numberof constraints is finite. In the particular case of linear problems theupper bound (or exact modulus) adopts a notably simplified expression. Ourapproach is based on variational techniques applied to certain difference ofconvex 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. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554])constitute the starting point of the present work.

Keywords

Convex semi-infinite programmingmodulus ofmetric regularityd.c. functions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 763 - 781
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burke, J.V. and Ferris, M.C., Weak sharp minima in mathematical programming. SIAM J. Contr. Opt. 31 (1993) 1340–1359. CrossRef
M.J. Cánovas, F.J. Gómez-Senent and J. Parra, On the Lipschitz modulus of the argmin mapping in linear semi-infinite optimization. Set-Valued Anal. (2007) Online First.
Cánovas, M.J., Klatte, D., López, M.A. and Parra, J., Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. 18 (2007) 717–732. CrossRef
M.J. Cánovas, A. Hantoute, M.A. López and J. Parra, Lipschitz behavior of convex semi-infinite optimization problems: A variational approach. J. Global Optim. 41 (2008) 1–13. CrossRef
M.J. Cánovas, A. Hantoute, M.A. López and J. Parra, Stability of indices in the KKT conditions and metric regularity in convex semi-infinite optimization. J. Optim. Theory Appl. (2008) Online First.
M.J. Cánovas, A. Hantoute, M.A. López and J. Parra, Lipschitz modulus of the optimal set mapping in convex semi-infinite optimization via minimal subproblems. Pacific J. Optim. (to appear).
E. De Giorgi, A. Marino and M. Tosques, 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.
Demyanov, V.F. and Rubinov, A.M., Quasidifferentiable functionals. Dokl. Akad. Nauk SSSR 250 (1980) 21–25 (in Russian).
V.F. Demyanov and A.M. Rubinov, Constructive nonsmooth analysis, Approximation & Optimization 7. Peter Lang, Frankfurt am Main (1995).
A.V. Fiacco and G.P. McCormick, Nonlinear programming. Wiley, New York (1968).
M.A. Goberna and M.A. López, Linear Semi-Infinite Optimization. John Wiley & Sons, Chichester, UK (1998).
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms, I. Fundamentals, Grundlehren der Mathematischen Wissenschaften 305. Springer-Verlag, Berlin (1993).
Ioffe, A.D., Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55 (2000) 103–162; English translation in Math. Surveys 55 (2000) 501–558. CrossRef
Ioffe, A.D., On rubustness of the regularity property of maps. Control Cybern. 32 (2003) 543–554.
D. Klatte and B. Kummer, Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer Academic Publ., Dordrecht (2002).
Klatte, D. and Kummer, B., Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16 (2005) 96–119. CrossRef
Klatte, D. and Thiere, G., A note of Lipschitz constants for solutions of linear inequalities and equations. Linear Algebra Appl. 244 (1996) 365–374. CrossRef
P.-J. Laurent, Approximation et Optimisation. Hermann, Paris (1972).
The, W. Li sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program. Linear Algebra Appl. 187 (1993) 15–40.
B.S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Springer-Verlag, Berlin (2006).
G. Nürnberger, Unicity in semi-infinite optimization, in Parametric Optimization and Approximation, B. Brosowski, F. Deutsch Eds., Birkhäuser, Basel (1984) 231–247.
Robinson, S.M., Bounds for error in the solution set of a perturbed linear program. Linear Algebra Appl. 6 (1973) 69–81. CrossRef
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, USA (1970).
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1997).
Studniarski, M. and Ward, D.E., Weak sharp minima: Characterizations and sufficient conditions. SIAM J. Contr. Opt. 38 (1999) 219–236. CrossRef
Valadier, M., Sous-différentiels d'une borne supérieure et d'une somme continue de fonctions convexes. C. R. Acad. Sci. Paris 268 (1969) 39–42.