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Set Intersection Theorems and Existence of Optimal Solutions

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Abstract

The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We consider asymptotic directions of a sequence of closed sets, and introduce associated notions of retractive, horizon, and critical directions, based on which we provide new conditions that guarantee the nonemptiness of the corresponding intersection. We show how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions, and to derive some new results, including a new extension of the Frank–Wolfe Theorem for (nonconvex) quadratic programming.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, M.I.T., Cambridge, MA, 02139, USA

    Dimitri P. Bertsekas

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Paul Tseng

Authors
  1. Dimitri P. Bertsekas
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  2. Paul Tseng
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Correspondence to Dimitri P. Bertsekas.

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Bertsekas, D.P., Tseng, P. Set Intersection Theorems and Existence of Optimal Solutions. Math. Program. 110, 287–314 (2007). https://doi.org/10.1007/s10107-006-0003-6

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  • DOI: https://doi.org/10.1007/s10107-006-0003-6

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