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, Volume 20, Issue 2, pp 310–327 | Cite as

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

  • M. J. Cánovas
  • M. A. López
  • B. S. Mordukhovich
  • J. Parra
Original Paper

Abstract

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel–Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system’s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system’s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

Keywords

Semi-infinite and infinite programming Parametric optimization Variational analysis Convex infinite inequality systems Quantitative stability Lipschitzian bounds Generalized differentiation Coderivatives Block perturbations 

Mathematics Subject Classification (2000)

90C34 90C25 49J52 49J53 65F22 

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References

  1. Cánovas MJ, Dontchev AL, López MA, Parra J (2005a) Metric regularity of semi-infinite constraint systems. Math Program 104:329–346 CrossRefGoogle Scholar
  2. Cánovas MJ, López MA, Parra J, Toledo FJ (2005b) Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems. Math Program 103:95–126 CrossRefGoogle Scholar
  3. Cánovas MJ, Gómez-Senent FJ, Parra J (2008) Regularity modulus of arbitrarily perturbed linear inequality systems. J Math Anal Appl 343:315–327 CrossRefGoogle Scholar
  4. Cánovas MJ, López MA, Mordukhovich BS, Parra J (2009) Variational analysis in semi-infinite and infinite programming, I: Stability of linear inequality systems of feasible solutions. SIAM J Optim 20:1504–1526 CrossRefGoogle Scholar
  5. Cánovas MJ, López MA, Mordukhovich BS, Parra J (2010a) Variational analysis in semi-infinite and infinite programming, II: Necessary optimality conditions. SIAM J Optim 20:2788–2806 CrossRefGoogle Scholar
  6. Cánovas MJ, López MA, Mordukhovich BS, Parra J (2010b) Quantitative stability and optimality conditions in convex semi-infinite and infinite programming, Research Report 14, Department of Mathematics, Wayne State University, Detroit, MI Google Scholar
  7. Cánovas MJ, López MA, Parra J, Toledo FJ (2011) Distance to ill-posedness for linear inequality systems under block perturbations. Convex and infinite dimensional cases. Optimization 60(7). doi:10.1080/02331934.2011.606624
  8. Deville R, Godefroy G, Zizler V (1993) Smoothness and renormings in Banach spaces. Longman, Harlow Google Scholar
  9. Dinh N, Goberna MA, López MA (2006) From linear to convex systems: Consistency, Farkas’ lemma and applications. J Convex Anal 13:279–290 Google Scholar
  10. Dinh N, Goberna MA, López MA (2010) On the stability of the feasible set in optimization problems. SIAM J Optim 20:2254–2280 CrossRefGoogle Scholar
  11. Dunford N, Schwartz JT (1988) Linear operators Part I: General theory. Wiley, New York Google Scholar
  12. Goberna MA, López MA (1998) Linear semi-infinite optimization. Wiley, Chichester Google Scholar
  13. Goberna MA, A López M, Todorov MI (1996) Stability theory for linear inequality systems. SIAM J Matrix Anal Appl 17:730–743 CrossRefGoogle Scholar
  14. Ioffe AD (2000) Metric regularity and subdifferential calculus. Russ Math Surv 55:501–558 CrossRefGoogle Scholar
  15. Ioffe AD (2010) On stability of solutions to systems of convex inequalities. Centre de Recerca Matemàtica, Preprint #984, November Google Scholar
  16. Ioffe AD, Sekiguchi Y (2009) Regularity estimates for convex multifunctions. Math Program 117:255–270 CrossRefGoogle Scholar
  17. Mordukhovich BS (1976) Maximum principle in problems of time optimal control with nonsmooth constraints. J Appl Math Mech 40:960–969 CrossRefGoogle Scholar
  18. Mordukhovich BS (1993) Complete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans Am Math Soc 340:1–35 CrossRefGoogle Scholar
  19. Mordukhovich BS (2006) Variational analysis and generalized differentiation, I: Basic theory, II: Applications. Springer, Berlin Google Scholar
  20. Phelps RR (1989) Convex functions, monotone operators and differentiability. Springer, Heidelberg CrossRefGoogle Scholar
  21. Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin CrossRefGoogle Scholar
  22. Schirotzek W (2007) Nonsmooth analysis. Springer, Berlin CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2011

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • M. A. López
    • 2
  • B. S. Mordukhovich
    • 3
  • J. Parra
    • 1
  1. 1.Center of Operations ResearchMiguel Hernández University of ElcheElche (Alicante)Spain
  2. 2.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain
  3. 3.Department of MathematicsWayne State UniversityDetroitUSA

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