Characterization of Radially Lower Semicontinuous Pseudoconvex Functions

Vsevolod I. Ivanov ()
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Vsevolod I. Ivanov: Technical University of Varna

Journal of Optimization Theory and Applications, 2020, vol. 184, issue 2, No 3, 368-383

Abstract: 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.

Keywords: Pseudoconvex function; Nonsmooth analysis; Nonsmooth optimization; Lower Dini directional derivative; 26B25; 49J52; 90C26 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10957-019-01604-w

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