Skip to main content

Advertisement

SpringerLink
Log in

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

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods and applications. SIAM Rev. 35, 380–429 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180(2), 491–518 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kirst, P., Stein, O.: Solving disjunctive optimization problems by generalized semi-infinite optimization techniques. J. Optim. Theory Appl. 169, 1079–1109 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahmed, F., Dür, M., Still, G.: Copositive programming via semi-infinite optimization. J. Optim. Theory Appl. 159, 322–349 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  10. Özöğür-Akyüz, S., Weber, G.-W.: Learning with infinitely many Kernels via semi-infinite programming. Optim. Methods Softw. 25(6), 937–970 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New-York (2000)

    Book  MATH  Google Scholar 

  12. Hettich, R., Still, G.: Second order optimality conditions for generalized semi-infinite programming problems. Optimization 34, 195–211 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rückmann, J.-J., Shapiro, A.: First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101, 677–691 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stein, O., Still, G.: On optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 104(2), 443–458 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  21. Stein, O.: Bi-Level Strategies in Semi-Infinite Programming. Kluwer, Boston (2003)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rückmann, J.-J.: On existence and uniqueness of stationary points in semi-infinite optimization. Math. Program. 86, 387–415 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Klatte, D.: On regularity and stability in semi-infinite optimization. Set-Valued Anal. 3, 101–111 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  30. Kruger, A., Minchenko, L., Outrat, J.: On relaxing the Mangasarian–Fromovitz constraint qualification. Positivity 18, 171–189 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kostyukova, O.I.: Optimality conditions for nonsmooth convex programming problems. Dokl. Natl. Akad. Nauk Belarusi 57(5), 22–27 (2013)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kostyukova, O.I., Tchemisova, T.V.: Optimality criteria without constraint qualifications for linear SDP problems. J. Math. Sci. 182(2), 126–143 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, L., Guo, F.: Semidefinite relaxations for semi-infinite polynomial programming. Comput. Optim. Appl. 58(1), 133–159 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, Surganov Str. 11, 220072, Minsk, Belarus

    Olga Kostyukova

  2. Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário Santiago, 3810-193, Aveiro, Portugal

    Tatiana Tchemisova

Authors
  1. Olga Kostyukova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana Tchemisova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatiana Tchemisova.

Additional information

Communicated by Juan Parra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kostyukova, O., Tchemisova, T. Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets. J Optim Theory Appl 175, 76–103 (2017). https://doi.org/10.1007/s10957-017-1150-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1150-z

Keywords

Mathematics Subject Classification

Search

Navigation