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

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Abstract

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.

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Authors and Affiliations

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

    O. I. Kostyukova

  2. Mathematical Department, University of Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    T. V. Tchemisova

  3. Department of Informatics, Belorussian State University of Informatics and Radio-electronics, P. Brovki street 6, Minsk, 220013, Belarus

    S. A. Yermalinskaya

Authors
  1. O. I. Kostyukova
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  2. T. V. Tchemisova
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  3. S. A. Yermalinskaya
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Corresponding author

Correspondence to T. V. Tchemisova.

Additional information

Communicated by F.A. Potra.

Research of the first author was partially supported by the state program of fundamental research Mathematical Models 13, Republic of Belarus. The second author was supported in part by FCT—Fundacão para Ciência e Tecnologia, Lisbon, Portugal.

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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, 325–342 (2010). https://doi.org/10.1007/s10957-009-9621-5

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  • DOI: https://doi.org/10.1007/s10957-009-9621-5

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