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Lopsided convergence: an extension and its quantification

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Abstract

Much of the development of lopsided convergence for bifunctions defined on product spaces was in response to motivating applications. A wider class of applications requires an extension that would allow for arbitrary domains, not only product spaces. This leads to an extension of the definition and its implications that include the convergence of solutions and optimal values of a broad class of minsup problems. In the process we relax the definition of lopsided convergence even for the classical situation of product spaces. We now capture applications in optimization under stochastic ambiguity, Generalized Nash games, and many others. We also introduce the lop-distance between bifunctions, which leads to the first quantification of lopsided convergence. This quantification facilitates the study of convergence rates of methods for solving a series of problems including minsup problems, Generalized Nash games, and various equilibrium problems.

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Notes

  1. We prefer “minsup problem” to “minimax problem” as the inner maximization may not be attained in much of our development.

  2. The need to check subsequences \(N\in \mathcal{N}_\infty ^{\scriptscriptstyle \#}\) is understood, but not explicitly stated in these references.

  3. Statements about convexity/concavity are the only ones that require a linear space in this paper.

  4. We deduce from a counterexample in [10] that the separability assumption cannot be relaxed.

  5. In parametric optimization and elsewhere sup-projections of bifunctions are sometimes called optimal value functions.

  6. Although the topology induced by the aw-distance is the same for any point selected, the value of the distance will depend on this choice; see [5, 27] and [26, Section 7.J] for details.

  7. A metric space is finitely compact if all its closed balls are compact.

  8. \(D:C\rightrightarrows Y\) is inner semicontinuous if for every \(x^\nu \in C \rightarrow x\in C\), \(\mathop {\mathrm{InnLim}}\nolimits D(x^\nu ) \supset D(x).\)

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Acknowledgements

This material is based upon work supported in part by DARPA under Grant HR0011-14-1-0060 and the U.S. Office of Naval Research under Grant No. N00014-17-2372.

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Authors and Affiliations

  1. Operations Research Department, Naval Postgraduate School, Monterey, CA, USA

    Johannes O. Royset

  2. Department of Mathematics, University of California, Davis, Davis, CA, USA

    Roger J-B Wets

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  1. Johannes O. Royset
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  2. Roger J-B Wets
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Correspondence to Johannes O. Royset.

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Royset, J.O., Wets, R.JB. Lopsided convergence: an extension and its quantification. Math. Program. 177, 395–423 (2019). https://doi.org/10.1007/s10107-018-1275-3

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