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Characterization of Radially Lower Semicontinuous Pseudoconvex Functions

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Abstract

In this paper, we prove that a radially lower semicontinuous function of one variable, defined on some interval, is pseudoconvex, if and only if its domain of definition can be split into three parts such that the function is strictly monotone decreasing without stationary points over the first subinterval, it is constant over the second one, and it is strictly monotone increasing without stationary points over the third subinterval. Each one or two of these parts may be empty or degenerate into a single point. The proof of this property is easy, when the function is differentiable. We consider functions, which are pseudoconvex with respect to the lower Dini directional derivative. This result follows from some known claims, but our theorem is a shot, directed to the target. We apply this characterization to obtain a complete characterization of strictly pseudoconvex functions. We also derive the respective results, when the function is radially lower semicontinuous in a real linear space. Several applications of the characterization are provided. A result due to Diewert, Avriel and Zang is extended to radially continuous functions.

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Acknowledgements

The author would like to express his gratitude to the anonymous referees for their helpful comments on the manuscript. This research is partially supported by the TU-Varna Grant No. 19/2019.

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  1. Department of Mathematics, Technical University of Varna, Varna, Bulgaria

    Vsevolod I. Ivanov

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  1. Vsevolod I. Ivanov
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Correspondence to Vsevolod I. Ivanov.

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Communicated by Constantin Zalinescu.

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Ivanov, V.I. Characterization of Radially Lower Semicontinuous Pseudoconvex Functions. J Optim Theory Appl 184, 368–383 (2020). https://doi.org/10.1007/s10957-019-01604-w

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