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

María J. Cánovas; Abderrahim Hantoute; Marco A. López; Juan Parra

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 4, page 763-781
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Cánovas, María J., et al. "Lipschitz modulus in convex semi-infinite optimization via d.c. functions." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2009): 763-781. <http://eudml.org/doc/245593>.

@article{Cánovas2009,
abstract = {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.},
author = {Cánovas, María J., Hantoute, Abderrahim, López, Marco A., Parra, Juan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {convex semi-infinite programming; modulus of metric regularity; d.c. functions},
language = {eng},
number = {4},
pages = {763-781},
publisher = {EDP-Sciences},
title = {Lipschitz modulus in convex semi-infinite optimization via d.c. functions},
url = {http://eudml.org/doc/245593},
volume = {15},
year = {2009},
}

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
PB - EDP-Sciences
VL - 15
IS - 4
SP - 763
EP - 781
AB - 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.
LA - eng
KW - convex semi-infinite programming; modulus of metric regularity; d.c. functions
UR - http://eudml.org/doc/245593
ER -

References

top
  1. [1] J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming. SIAM J. Contr. Opt. 31 (1993) 1340–1359. Zbl0791.90040MR1234006
  2. [2] 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. Zbl1156.90448MR2465504
  3. [3] M.J. Cánovas, D. Klatte, M.A. López and J. Parra, Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. 18 (2007) 717–732. Zbl1211.90256MR2345965
  4. [4] 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. Zbl1190.90247MR2386592
  5. [5] 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. Zbl1190.90246MR2453333
  6. [6] 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). Zbl1162.49024
  7. [7] 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. Zbl0465.47041MR636814
  8. [8] V.F. Demyanov and A.M. Rubinov, Quasidifferentiable functionals. Dokl. Akad. Nauk SSSR 250 (1980) 21–25 (in Russian). Zbl0456.49016MR556111
  9. [9] V.F. Demyanov and A.M. Rubinov, Constructive nonsmooth analysis, Approximation & Optimization 7. Peter Lang, Frankfurt am Main (1995). Zbl0887.49014MR1325923
  10. [10] A.V. Fiacco and G.P. McCormick, Nonlinear programming. Wiley, New York (1968). Zbl0193.18805MR243831
  11. [11] M.A. Goberna and M.A. López, Linear Semi-Infinite Optimization. John Wiley & Sons, Chichester, UK (1998). Zbl0909.90257MR1628195
  12. [12] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms, I. Fundamentals, Grundlehren der Mathematischen Wissenschaften 305. Springer-Verlag, Berlin (1993). Zbl0795.49001MR1261420
  13. [13] A.D. Ioffe, Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55 (2000) 103–162; English translation in Math. Surveys 55 (2000) 501–558. Zbl0979.49017MR1777352
  14. [14] A.D. Ioffe, On rubustness of the regularity property of maps. Control Cybern. 32 (2003) 543–554. Zbl1127.49023
  15. [15] D. Klatte and B. Kummer, Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer Academic Publ., Dordrecht (2002). Zbl1173.49300MR1909427
  16. [16] D. Klatte and B. Kummer, Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16 (2005) 96–119. Zbl1097.90058MR2177771
  17. [17] D. Klatte and G. Thiere, A note of Lipschitz constants for solutions of linear inequalities and equations. Linear Algebra Appl. 244 (1996) 365–374. Zbl0860.15015MR1403290
  18. [18] P.-J. Laurent, Approximation et Optimisation. Hermann, Paris (1972). Zbl0238.90058MR467080
  19. [19] W. Li, The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program. Linear Algebra Appl. 187 (1993) 15–40. Zbl0809.65057MR1221694
  20. [20] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Springer-Verlag, Berlin (2006). Zbl1100.49002MR2191744
  21. [21] G. Nürnberger, Unicity in semi-infinite optimization, in Parametric Optimization and Approximation, B. Brosowski, F. Deutsch Eds., Birkhäuser, Basel (1984) 231–247. Zbl0586.90065MR882207
  22. [22] S.M. Robinson, Bounds for error in the solution set of a perturbed linear program. Linear Algebra Appl. 6 (1973) 69–81. Zbl0283.90028MR317760
  23. [23] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, USA (1970). Zbl0193.18401MR274683
  24. [24] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1997). Zbl0888.49001MR1491362
  25. [25] M. Studniarski and D.E. Ward, Weak sharp minima: Characterizations and sufficient conditions. SIAM J. Contr. Opt. 38 (1999) 219–236. Zbl0946.49011MR1740599
  26. [26] M. Valadier, Sous-différentiels d’une borne supérieure et d’une somme continue de fonctions convexes. C. R. Acad. Sci. Paris 268 (1969) 39–42. Zbl0164.43302MR241975

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.