Existence theorem and optimality conditions for a class of convex semi-infinite problems with noncompact index sets

  • Olga Kostyukova Institute of Mathematics, Belarusian National Academy of Sciences
  • Tatiana Tchemisova Mathematics Department, University of Aveiro
  • Maryia Kurdina Institute of Mathematics, Belarusian National Academy of Sciences
DOI: https://doi.org/10.19139/soic.v5i4.362
Keywords: Semi-Infinite Programming (SIP), Linear Programming (LP), Quadratic Programming (QP),

Abstract

The paper is devoted to study of a special class of semi-infinite problems arising in nonlinear parametric Semi-infinite Programming, when the differential properties of the solutions are being studied. These problems are convex and possess noncompact index sets. In the paper, we present conditions guaranteeing the existence of optimal solutions, and prove new optimality criterion. An example illustrating the obtained results is presented.

References

Asprey S.P., Maccietto S. Eds. Dynamic model development: methods, theory and applications, In: Proceedings of the Workshop on the Life af a Process Model-From Conception to Action, Imperial Colledge, London, UK, October 25-26, 2000.

Bonnans J.F., Shapiro A. Perturbation analysis of optimization problems, Springer-Verlag, New-York, 2000.

Canovas M. J., Lopez M. A., Mordukhovich B. S. and Parra J. Variational analysis in semi-infinite and infinite programming, II: necessary optimality conditions, SIAM Journal on Optimization, vol. 20, no. 6, pp. 2788-2806, 2010.

Eaves B.C. On Quadratic Programming, Management Science, Theory Series, vol. 17, no. 11, pp. 698–711, 1971.

Goberna M.A., and Lopez M. A. Linear Semi-Infinite Optimization, Wiley, Chichester, 1998.

Goberna M.A., Lopes M.A. (eds.) Semi-Infinite Programming: recent advances, Kluwer, Dordrecht, 2001.

Hettich R., Jongen H.Th. Semi-Infinite Programming: conditions of optimality and applications, in: J. Stoer, ed., Optimization Techniques, Part 2, Lecture Notes in Control and Information Sciences, no. 7, pp. 1–11, 1978.

Hettich R., Kortanek K.O. Semi-Infinite Programming: theory, methods and applications, SIAM Rev., vol. 35, pp. 380–429, 1993.

Hettich R., Still G. Second order optimality conditions for generalized semi-infinite programming problems, Optimization, vol. 34, pp. 195–211, 1995.

Kostyukova O.I., Tchemisova T.V., Kurdina M.A. A study of one class of NLP problems arising in parametric Semi-Infinite Programming, Optimization Methods and Software, Vol. 32 , Iss. 6, pp.1218-1243, 2017.

Kostyukova O.I., Tchemisova T.V. Implicit optimality criterion for convex SIP problem with box constrained index set, TOP, vol. 20, no. 2, pp. 475–502, 2012.

Kostyukova O.I., Tchemisova T.V., and Kurdina M.A. On Optimal Properties of Special Nonlinear and Semi-infinite Problems Arising in Parametric Optimization, Stat., Optim. Inf. Comput., vol. 5, pp. 99-108, 2017.

Mordukhovich B., Nghia T. T. A. Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs, Math. Program., Ser. B, vol. 139, pp. 271-300, 2013. DOI 10.1007/s10107-013-0672-x.

Stein O., Still G. On optimality conditions for generalized semi-infinite programming problems, J. Optim. Theory Appl., vol. 104, no. 2, pp. 443–458, 2000.

Wang L., Guo F. Semidefinite relaxations for semi-infinite polynomial programming, Computational Optimization and Applications, vol. 58, no. 1, pp. 133–159, 2013.

Weber G.-W., Kropat E., Alparslan G¨ok 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.

Published
2017-11-30
How to Cite
Kostyukova, O., Tchemisova, T., & Kurdina, M. (2017). Existence theorem and optimality conditions for a class of convex semi-infinite problems with noncompact index sets. Statistics, Optimization & Information Computing, 5(4), 278-294. https://doi.org/10.19139/soic.v5i4.362
Issue
Vol 5 No 4 (2017)
Section
Research Articles
Authors who publish with this journal agree to the following terms:

    1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
    2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
    3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).