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Characterizing the solution set of convex optimization problems without convexity of constraints

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Abstract

Some characterizations of solution sets of a convex optimization problem with a convex feasible set described by tangentially convex constraints are given. The results are expressed in terms of convex subdifferentials, tangential subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian-type function and establish a constant pseudo Lagrangian-type property for the solution set. This property is still valid in the case of a pseudoconvex locally Lipschitz objective function, and then used to derive Lagrange multiplier-based characterizations of the solution set. Some examples are given to illustrate the significances of our theoretical results.

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Notes

  1. A tangentially convex function \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) at \(x \in {\mathbb {R}}^n\) is said to be pseudoconvex atx (see [26]) if \( \forall y\in {\mathbb {R}}^n,\; f'(x,y-x)\ge 0 \Longrightarrow f(y)\ge f(x). \)

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Acknowledgements

The authors would like to express their sincere thanks to anonymous referees for helpful suggestions and valuable comments for the paper. This research was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0026/2555) and the Thailand Research Fund, Grant No. RSA6080077.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Nithirat Sisarat & Rabian Wangkeeree

  2. Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok, Thailand

    Rabian Wangkeeree

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  1. Nithirat Sisarat
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  2. Rabian Wangkeeree
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Correspondence to Rabian Wangkeeree.

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Sisarat, N., Wangkeeree, R. Characterizing the solution set of convex optimization problems without convexity of constraints. Optim Lett 14, 1127–1144 (2020). https://doi.org/10.1007/s11590-019-01397-x

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