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Lagrange Multiplier Characterizations of Constrained Best Approximation with Infinite Constraints

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Abstract

In this paper, we first employ the subdifferential closedness condition and Guignard’s constraint qualification to present “dual cone characterizations” of the constraint set \( \varOmega \) with infinite nonconvex inequality constraints, where the constraint functions are Fréchet differentiable that are not necessarily convex. We next provide sufficient conditions for which the “strong conical hull intersection property” (strong CHIP) holds, and moreover, we establish necessary and sufficient conditions for characterizing “perturbation property” of the best approximation to any \(x \in {\mathcal {H}}\) from the convex set \( \tilde{\varOmega }:=C \cap \varOmega \) by using the strong CHIP of \(\lbrace C,\varOmega \rbrace ,\) where C is a non-empty closed convex set in the Hilbert space \({\mathcal {H}}.\) Finally, we derive the “Lagrange multiplier characterizations” of constrained best approximation under the subdifferential closedness condition and Guignard’s constraint qualification. Several illustrative examples are presented to clarify our results.

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Acknowledgements

The authors are very grateful to the two anonymous referees and the Associate Editor for their valuable comments regarding an earlier version of this paper. The comments of the referees and the Associate Editor were very useful and they helped us to improve the paper significantly. This research was partially supported by Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Iran, Grant No. 97/3267.

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Authors and Affiliations

  1. Department of Mathematics, Shahid Bahonar University of Kerman, 7616914111, Kerman, Iran

    Hassan Bakhtiari & Hossein Mohebi

  2. Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, 7616914111, Kerman, Iran

    Hossein Mohebi

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  1. Hassan Bakhtiari
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  2. Hossein Mohebi
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Correspondence to Hossein Mohebi.

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Communicated by Marco Antonio López-Cerdá.

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Bakhtiari, H., Mohebi, H. Lagrange Multiplier Characterizations of Constrained Best Approximation with Infinite Constraints. J Optim Theory Appl 189, 814–835 (2021). https://doi.org/10.1007/s10957-021-01856-5

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