Convex Semi-Infinite Programming: Implicit Optimality Criterion Based on the Concept of Immobile Indices

  • O. I. Kostyukova
  • T. V. Tchemisova
  • S. A. Yermalinskaya


We state a new implicit optimality criterion for convex semi-infinite programming (SIP) problems. This criterion does not require any constraint qualification and is based on concepts of immobile index and immobility order. Given a convex SIP problem with a continuum of constraints, we use an information about its immobile indices to construct a nonlinear programming (NLP) problem of a special form. We prove that a feasible point of the original infinite SIP problem is optimal if and only if it is optimal in the corresponding finite NLP problem. This fact allows us to obtain new efficient optimality conditions for convex SIP problems using known results of the optimality theory of NLP. To construct the NLP problem, we use the DIO algorithm. A comparison of the optimality conditions obtained in the paper with known results is provided.

Convex semi-infinite programming Nonlinear programming Optimality criteria Constraint qualifications 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Goberna, M.A., Lopez, M.A.: Linear Semi-infinite Optimization. Wiley, New York (1998) MATHGoogle Scholar
  2. 2.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods and applications. SIAM Rev. 35, 380–429 (1993) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Polak, E.: Semi-infinite optimization in engineering design. In: Fiacco, A.V., Kortanek, K.O. (eds.) Semi-infinite Programming and Applications. Lecture Notes in Economics and Mathematical Systems, pp. 236–248. Springer, New York (1983) Google Scholar
  4. 4.
    Still, G.: Generalized semi-infinite programming: theory and methods. Eur. J. Oper. Res. 119, 301–313 (1999) MATHCrossRefGoogle Scholar
  5. 5.
    Ben Tal, A., Kerzner, L., Zlobec, S.: Optimality conditions for convex semi-infinite programming problems. Naval Res. Log. 27(3), 413–435 (1980) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14(2), 534–547 (2003) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Rückmann, J.J., Shapiro, A.: First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101(3), 677–691 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Klatte, D.: Stable local minimizers in semi-infinite optimization: regularity and second-order conditions. J. Comput. Appl. Math. 56(1–2), 137–157 (1994) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hettich, R., Still, G.: Second order optimality conditions for generalized semi-infinite programming problems. Optimization 34, 195–211 (1995) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goberna, M.A., Jeyakumar, V., Lópes, M.A.: Necessary and sufficient constraint qualifications for systems of infinite convex inequalities. J. Nonlinear Anal. 68, 1184–1194 (2008) MATHCrossRefGoogle Scholar
  11. 11.
    Li, W., Nahak, Ch., Singer, I.: Constraint qualifications for semi-infinite systems of convex inequalities. SIAM J. Optim. 11(1), 31–52 (2000) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hettich, R., Jongen, H.T.: On first and second order conditions for local optima for optimization problems in finite dimensions. Methods Oper. Res. 23, 82–97 (1977) MathSciNetGoogle Scholar
  13. 13.
    Arutiunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer Academic, Dordrecht (2000) Google Scholar
  14. 14.
    Brezhneva, O.A., Tret’yakov, A.: Optimality conditions for degenerate extremum problems with equality constraints. SIAM J. Control Optim. 42(2), 729–745 (2003) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Izmailov, A.F., Solodov, M.V.: On optimality conditions of cone-constrained optimization. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, pp. 3162–3167 (2002) Google Scholar
  16. 16.
    Kortanek, K.O., Medvedev, V.G.: Semi-infinite programming and new applications in finance. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization. Kluwer Academic, Dordrecht (2005) Google Scholar
  17. 17.
    Kostyukova, O.I.: Investigation of the linear extremal problems with continuum constraints. Report No. 26/336, Institute of Mathematics of Belorussian Academy of Sciences (1988) Google Scholar
  18. 18.
    Kostyukova, O.I., Tchemisova, T.V., Yermalinskaya, S.A.: On the algorithm of determination of immobile indices for convex SIP problems. Int. J. Appl. Math. Stat. 13(8), 13–33 (2008) MATHMathSciNetGoogle Scholar
  19. 19.
    Jongen, H.T., Wetterling, W., Zwier, G.: On sufficient conditions for local optimality in semi-infinite programming. Optimization 18(2), 165–178 (1987) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fajardo, M.D., Lopes, M.A.: Some results about the facial geometry of convex semi-infinite systems. Optimization 55(5–6), 661–684 (2006) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) MATHGoogle Scholar
  22. 22.
    Stein, O., Still, G.: On optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 104(2), 443–458 (2000) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Jongen, H.Th., Rückmann, J.J., Stein, O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83, 145–158 (1998) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • O. I. Kostyukova
    • 1
  • T. V. Tchemisova
    • 2
  • S. A. Yermalinskaya
    • 3
  1. 1.Institute of Mathematics, National Academy of Sciences of BelarusMinskBelarus
  2. 2.Mathematical DepartmentUniversity of AveiroAveiroPortugal
  3. 3.Department of InformaticsBelorussian State University of Informatics and Radio-electronicsMinskBelarus

Personalised recommendations