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Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function

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Abstract

The strict lower semicontinuity property (slsc property) of the level sets of a real-valued functionf defined on a subsetCR n was introduced by Zang, Choo, and Avriel (Ref. 1). They showed a class of functions for which the slsc property is equivalent to invexity, i.e., the statement that every stationary point off overC is a global minimum. In this paper, we study the relationship between the slsc property of the level sets and invexity for another class of functions. Namely, we consider the class formed by all locally Lipschitz real-valued functions defined on an open set Ω containingC. For these functions, invexity implies the slsc property of the level sets, but not conversely.

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Communicated by M. Avriel

The authors would like to thank Dr. B. D. Craven and the referees for helpful comments and suggestions.

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Phuong, T.D., Sach, P.H. & Yen, N.D. Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function. J Optim Theory Appl 87, 579–594 (1995). https://doi.org/10.1007/BF02192135

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