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Approximations of Optimization-Related Problems in Terms of Variational Convergence

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Abstract

In this paper, we first adjust the known definition of epi/hypo convergence of bivariate extended-real-valued functions for finite-valued ones defined on product sets. Then, we develop basic variational properties of this type of convergence. Using these results, we consider approximations of optimization-related problems in terms of epi/hypo convergence and related types of convergence, for some selected important cases: equilibrium problems, multiobjective optimization, and Nash equilibria. Our approximation results can be viewed also as contributions to qualitative stability (as the terminology “stability” was also used instead of “approximation” for similar results in the literature).

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Acknowledgements

This research was supported by Vietnam National University Hochiminh City (VNU-HCM) under grant number B2015-28-03. The second author is very grateful to Professor Roger J.-B. Wets for the suggestion of the topic and many important discussions during his work. He thanks also Professor Alejandro Jofré for helpful discussions. The authors are deeply indebted to the two anonymous referees for their valuable remarks and suggestions, which have helped them much in completing the final version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, College of Can Tho, Can Tho, Vietnam

    Huynh Thi Hong Diem

  2. Department of Mathematics, International University, Vietnam National University Ho Chi Minh City, Linh Trung, Thu Duc, Ho Chi Minh City, Vietnam

    Phan Quoc Khanh

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  1. Huynh Thi Hong Diem
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  2. Phan Quoc Khanh
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Correspondence to Phan Quoc Khanh.

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Professor Phan Quoc Khanh was Plenary speaker at the Vietnam Congress of Mathematicians 2013.

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Hong Diem, H.T., Khanh, P.Q. Approximations of Optimization-Related Problems in Terms of Variational Convergence. Vietnam J. Math. 44, 399–417 (2016). https://doi.org/10.1007/s10013-015-0148-9

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