Stability of Linear Inequality Systems in a Parametric Setting

  • M. J. Cánovas
  • M. A. López
  • J. Parra


In this paper, we propose a parametric approach to the stability theory for the solution set of a semi-infinite linear inequality system in the n-dimensional Euclidean space \({\mathbb R}^{n}\). The main feature of this approach is that the coefficient perturbations are modeled through the so-called mapping of parametrized systems, which assigns to each parameter, ranging in a metric space, a subset of \({\mathbb R}^{n+1}\). Each vector of this image set provides the coefficients of an inequality in \({\mathbb R}^{n}\) and the whole image set defines the inequality system associated with the parameter. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. The paper is focused mainly on the structural stability of the feasible set mapping, providing a characterization of the Berge lower-semicontinuity property. The role played by the strong Slater qualification is analyzed in detail.


Stability parametrized systems linear inequality systems feasible set mapping 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zlobec, S. 2001Stability in Linear Programming Models: An Index Set ApproachAnnals of Operations Research101363382Google Scholar
  2. 2.
    Zlobec, S. 2001Fritx John Conditions, Encyclopedia of Mathematics, Supplement IIIKluwer Academic PublishersDordrecht, Holland164166Google Scholar
  3. 3.
    Schochetman, I.E. 1990Pointwise Version of the Maximum Theorem with Applications in OptimizationApplied Mathematics Letters38992Google Scholar
  4. 4.
    Cánovas, M.J., López, M.A., Parra, J., Todorov, M.I. 1999Stability and Well-Posedness in Linear Semi-Infinite ProgrammingSIAM Journal on Optimization108298Google Scholar
  5. 5.
    Robinson, S.M. 1975Stability Theory for Systems of Inequalities, Part 1: Linear SystemsSIAM Journal on Numerical Analysis12754772Google Scholar
  6. 6.
    Cánovas, M.J., López, M.A., Parra, J. 2002Upper Semicontinuity of the Feasible Set Mapping for Linear Inequality SystemsSet-Valued Analysis10361378Google Scholar
  7. 7.
    Goberna, M.A., López, M.A. 1996Topological Stability of Linear Semi-Infinite Inequality SystemsJournal on Optimization Theory and Applications89227236Google Scholar
  8. 8.
    Goberna, M.A., López, M.A., Todorov, M.I. 1996Stability Theory for Linear Inequality SystemsSIAM Journal on Matrix Analysis and Applications17730743Google Scholar
  9. 9.
    Goberna, M.A., López, M.A., Todorov, M.I. 1997Stability Theory for Linear Inequality Systems, Part 2: Upper Semicontinuity of the Solution Set MappingSIAM Journal on Optimization711381151Google Scholar
  10. 10.
    Brosowski, B. 1984Parametric Semi-Infinite Linear Programming, Part 1: Continuity of the Feasible Set and of the Optimal ValueMathematical Programming Study211842Google Scholar
  11. 11.
    Fischer, T. 1983

    Contributions to Semi-Infinite Linear Optimization

    Brosowski, B.Martensen, E. eds. Approximation and Optimization in Mathematical PhysicsPeter LangFrankfurt-am-Main, Germany175199
    Google Scholar
  12. 12.
    Cánovas, M.J., López, M.A., Parra, J. 2002Stability of the Feasible Set and Linear Inequality Systems: A Carrier Index Set ApproachLinear Algebra and Its Applications35783105Google Scholar
  13. 13.
    Cánovas M.J., López M.A. and Parra J. On the Equivalence of Parametric Contexts for Linear Inequality Systems, Working Paper, Operations Research Center, Miguel Hernández University of Elche, 2004.Google Scholar
  14. 14.
    Rockafellar, R.T., Wets, R.J.B. 1998Variational AnalysisSpringer VerlagBerlin, GermanyGoogle Scholar
  15. 15.
    Beer, G. 1993Topologies on Closed and Closed Convex SetsKluwer Academic PublishersDordrecht HollandGoogle Scholar
  16. 16.
    Yong-jin, Z. 1966Generalizations of Some Fundamental Theorems on Linear InequalitiesActa Mathematica Sinica162540Google Scholar
  17. 17.
    Goberna, M.A., López, M.A. 1998Linear Semi-Infinite OptimizationJohn Wiley and SonsChichester, UKGoogle Scholar
  18. 18.
    Psenichnyj, B.N. 1966Linear Optimal Control ProblemsSIAM Journal of Control4577593Google Scholar
  19. 19.
    Psenichnyj B.N., Necessary Optimality Conditions, Teubner, Leipzig, Germany, 1972 (in German).Google Scholar
  20. 20.
    Hettich R. and Jongen H.T., Semi-Infinite Programming: Conditions of Optimality and Applications, Proceedings of the 8th IFIP Conference on Optimization Techniques, Edited by J. Stoer, Springer Verlag. New York, NY, pp. 1–11, (1978)Google Scholar
  21. 21.
    Jongen, H.T., Wetterling, W., Zwier, G. 1987On Sufficient Conditions for Local Optimality and ApplicationsOptimization18165178Google Scholar
  22. 22.
    Shapiro, A. 1985Second-Order Derivatives of External-Value Functions and Optimality Conditions for Semi-Infinite ProgramsMathematical of Operations Research10207219Google Scholar
  23. 23.
    Klatte, D. 1992

    Stability of Stationary Solutions in Semi-Infinite Optimization via the Reduction Approach

    Pallaschke, D. eds. Proceedings of the French-German Conference on Optimization.Springer VerlagBerlin Germany155170
    Google Scholar
  24. 24.
    Valadier, M.M. 1969Sous-Différentiels d’une Borne Supérieure et d’une Somme Continue de Fonctions ConvexesComptes Rendus de l’Academie des Sciences de Paris, Série A2683942Google Scholar
  25. 25.
    Hiriart Urruty, J.B., Lemarechal, C. 1991Convex Analysis and Minimization Algorithms, ISpringer VerlagBerlin, GermanyGoogle Scholar
  26. 26.
    Volle, M. 1993Sous-Différentiel d’une Envelope Supérieure des Fonctions ConvexesComptes Rendus de l’Academie des Sciences de Paris, Série I317845849Google Scholar
  27. 27.
    Rockafellar, R.T. 1970Convex AnalysisPriceton University PressPrinceton, New JerseyGoogle Scholar
  28. 28.
    Goberna M. A., López M. A., and TODOROV, M. I., A Sup-Function Approach to Linear Semi-Infinite Optimization, Journal of Mathematical Sciences, 116, pp. 3359–3368, (2003)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • M. A. López
    • 2
  • J. Parra
    • 1
  1. 1.Associate Professor, Operations Research CenterMiguel Hernández University of ElcheAlicanteSpain
  2. 2.Professor, Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain

Personalised recommendations