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Farkas’ lemma: three decades of generalizations for mathematical optimization

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Abstract

In this paper we present a survey of generalizations of the celebrated Farkas’s lemma, starting from systems of linear inequalities to a broad variety of non-linear systems. We focus on the generalizations which are targeted towards applications in continuous optimization. We also briefly describe the main applications of generalized Farkas’ lemmas to continuous optimization problems.

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Notes

  1. In Dinh et al. (2013b), it was proved that Theorem 7.4 implies approximate Hahn–Banach theorem but in fact, the two theorems are equivalent.

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Authors and Affiliations

  1. Department of Mathematics, International University Vietnam National University-Ho Chi Minh City, Ho Chi Minh City, Vietnam

    N. Dinh

  2. Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia

    V. Jeyakumar

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  1. N. Dinh
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  2. V. Jeyakumar
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Correspondence to N. Dinh.

Additional information

This invited paper is discussed in the comments available at doi:10.1007/s11750-014-0315-2; doi:10.1007/s11750-014-0316-1; doi:10.1007/s11750-014-0317-0; doi:10.1007/s11750-014-0318-z.

Research was partially supported by grants from the Australian Research Council and from NAFOSTED, Vietnam.

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Dinh, N., Jeyakumar, V. Farkas’ lemma: three decades of generalizations for mathematical optimization. TOP 22, 1–22 (2014). https://doi.org/10.1007/s11750-014-0319-y

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