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Zero duality gaps in infinite-dimensional programming

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Abstract

In this paper we study the following infinite-dimensional programming problem: (P) μ≔inff 0(x), subject tox∈C,f i(x)≤0,i∈I, whereI is an index set with possibly infinite cardinality andC is an infinite-dimensional set. Zero duality gap results are presented under suitable regularity hypotheses for convex-like (nonconvex) and convex infinitely constrained program (P). Various properties of the value function of the convex-like program and its connections to the regularity hypotheses are studied. Relationships between the zero duality gap property, semicontinuity, and ε-subdifferentiability of the value function are examined. In particular, a characterization for a zero duality gap is given, using the ε-subdifferential of the value function without convexity.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of New South Wales, Kensington, New South Wales, Australia

    V. Jeyakumar (Lecturer)

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

    H. Wolkowicz (Associate Professor)

Authors
  1. V. Jeyakumar
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  2. H. Wolkowicz
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Additional information

Communicated by A. V. Fiacco

The authors are extremely grateful to the referees for their constructive criticisms and helpful suggestions which have contributed to the final preparation of this paper. This research was partially completed while the first author was a visitor of the Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.

This work was partially supported by NSERC Grant No. A9161.

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Jeyakumar, V., Wolkowicz, H. Zero duality gaps in infinite-dimensional programming. J Optim Theory Appl 67, 87–108 (1990). https://doi.org/10.1007/BF00939737

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