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Journal of Optimization Theory and Applications

, Volume 175, Issue 1, pp 76–103 | Cite as

Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets

  • Olga Kostyukova
  • Tatiana TchemisovaEmail author
Article

Abstract

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems.

Keywords

Convex programming Semi-infinite programming (SIP) Nonlinear programming (NLP) Convex set Finitely representable set Constraint qualifications (CQ) Immobile index Optimality conditions 

Mathematics Subject Classification

90C25 90C30 90C34 

Notes

Acknowledgements

The authors are sincerely grateful to two anonymous reviewers for their valuable comments, suggestions, and corrections that allowed to improve the presentation of the results. Our special thanks to the Editors for the useful advices that helped to prepare the revised version. This work was partially supported by the state research program “Convergence” of Republic Belarus: Task 1.3.0 “Development of the methods for solving the uncorrect problems of the control theory for distributed dynamical systems”, and by Portuguese funds through CIDMA—Center for Research and Development in Mathematics and Applications, and FCT—Portuguese Foundation for Science and Technology, within the Project UID/MAT/04106/2013.

References

  1. 1.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods and applications. SIAM Rev. 35, 380–429 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180(2), 491–518 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kirst, P., Stein, O.: Solving disjunctive optimization problems by generalized semi-infinite optimization techniques. J. Optim. Theory Appl. 169, 1079–1109 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ahmed, F., Dür, M., Still, G.: Copositive programming via semi-infinite optimization. J. Optim. Theory Appl. 159, 322–349 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Weber, G.-W., Kropat, E., Alparslan Gök, S.Z.: Semi-infinite and conic optimization in modern human life and financial sciences under uncertainty. In: ISI Proceedings of 20th Mini-EURO Conference, pp. 180–185 (2008)Google Scholar
  6. 6.
    Mehrotra, S., Papp, D.: A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization. SIAM J. Optim. 24(4), 1670–1697 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sachs, E.W.: Semi-infinite programming in control. In: Reemtsen, R., Rückmann, J.-J. (eds.) Semi-Infinite Programming, pp. 389–411. Springer, Boston (1998)CrossRefGoogle Scholar
  8. 8.
    Weber, G.-W.: Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization. J. Stat. Manag. Syst. 5, 359–388 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kropat, E., Pickl, S., Rössler, A., Weber, G.-W.: A new algorithm from semi-infinite optimization for a problem of time-minimal control. Vycislitel’nye technologii (Comput. Technol.) 5(4), 67–81 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Özöğür-Akyüz, S., Weber, G.-W.: Learning with infinitely many Kernels via semi-infinite programming. Optim. Methods Softw. 25(6), 937–970 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New-York (2000)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hettich, R., Still, G.: Second order optimality conditions for generalized semi-infinite programming problems. Optimization 34, 195–211 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rückmann, J.-J., Shapiro, A.: First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101, 677–691 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Klatte, D.: Stable local minimizers in semi-infinite optimization: regularity and second-order conditions. J. Comput. Appl. Math. 56(1–2), 137–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stein, O., Still, G.: On optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 104(2), 443–458 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yang, X., Chen, Z., Zhou, J.: Optimality conditions for semi-infinite and generalized semi-infinite programs via lower order exact penalty functions. J. Optim. Theory Appl. 169(3), 984–1012 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kostyukova, O.I., Tchemisova, T.V.: Implicit optimality criterion for convex SIP problem with box constrained index set. TOP 20(2), 475–502 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, C., Ng, K.F., Pong, T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19(1), 163–187 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fajardo, M.D., López, M.A.: Some results about the facial geometry of convex semi-infinite systems. Optimization 55(5–6), 661–684 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cánovas, M.J., López, M.A, Mordukhovich, B.S., Parra, J.: Variational Analysis in Semi-Infinite and Infinite Programming, II: Necessary Optimality Conditions, Research Report MI 48202. Wayne State University (2009)Google Scholar
  21. 21.
    Stein, O.: Bi-Level Strategies in Semi-Infinite Programming. Kluwer, Boston (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Jongen, H.T., Twilt, F., Weber, G.-W.: Semi-infinite optimization: structure and stability of the feasible set. J. Optim. Theory Appl. 72, 529–552 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rückmann, J.-J.: On existence and uniqueness of stationary points in semi-infinite optimization. Math. Program. 86, 387–415 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Klatte, D.: On regularity and stability in semi-infinite optimization. Set-Valued Anal. 3, 101–111 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kostyukova, O.I., Tchemisova, T.V.: On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets. Optimization 63(1), 67–91 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kostyukova, O.I., Tchemisova, T.V., Yermalinskaya, S.A.: Convex semi-infinite programming: implicit optimality criterion based on the concept of immobile indices. J. Optim. Theory Appl. 145(2), 325–342 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kostyukova, O.I., Tchemisova, T.V.: Convex SIP problems with finitely representable compact index sets: immobile indices and the properties of an auxiliary NLP problem. Set-Valued Var. Anal. 23(3), 519–546 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kostyukova, O.I., Tchemisova, T.V.: A Constructive Algorithm of Determination of the Sets of Immobile Indices in Convex SIP Problems with Polyhedral Index Sets. Working paper of the University of Aveiro (2012)Google Scholar
  29. 29.
    Levin, V.L.: Application of E. Helly’s theorem to convex programming, problems of the best approximation and related questions. Math. USSR Sb. 8(2), 235–247 (1969)CrossRefGoogle Scholar
  30. 30.
    Kruger, A., Minchenko, L., Outrat, J.: On relaxing the Mangasarian–Fromovitz constraint qualification. Positivity 18, 171–189 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kostyukova, O.I.: Optimality conditions for nonsmooth convex programming problems. Dokl. Natl. Akad. Nauk Belarusi 57(5), 22–27 (2013)MathSciNetGoogle Scholar
  32. 32.
    Bomze, I.M., Dür, M., de Klerk, E., Roos, C., Quist, A.J., Terlaky, T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18(4), 301–320 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kostyukova, O.I., Tchemisova, T.V.: Optimality criteria without constraint qualifications for linear SDP problems. J. Math. Sci. 182(2), 126–143 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, L., Guo, F.: Semidefinite relaxations for semi-infinite polynomial programming. Comput. Optim. Appl. 58(1), 133–159 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)zbMATHGoogle Scholar
  36. 36.
    Mordukhovich, B., Nghia, T.T.A.: Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs. Math. Program. 139(1–2), 271–300 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Parpas, P., Rustem, B.: An algorithm for the global optimization of a class of continuous minimax problems. J. Optim. Theory Appl. 141(2), 461–473 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of MathematicsNational Academy of Sciences of BelarusMinskBelarus
  2. 2.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal

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