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Characterization of Radially Lower Semicontinuous Pseudoconvex Functions

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  • Vsevolod I. Ivanov

    (Technical University of Varna)

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.

Suggested Citation

  • Vsevolod I. Ivanov, 2020. "Characterization of Radially Lower Semicontinuous Pseudoconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 368-383, February.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:2:d:10.1007_s10957-019-01604-w
    DOI: 10.1007/s10957-019-01604-w
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    References listed on IDEAS

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    3. Satoshi Suzuki & Daishi Kuroiwa, 2015. "Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 62(3), pages 431-441, July.
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    6. Vsevolod I. Ivanov, 2013. "Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 65-84, July.
    7. Diewert, W. E. & Avriel, M. & Zang, I., 1981. "Nine kinds of quasiconcavity and concavity," Journal of Economic Theory, Elsevier, vol. 25(3), pages 397-420, December.
    8. Vsevolod Ivanov, 2013. "Characterizations of pseudoconvex functions and semistrictly quasiconvex ones," Journal of Global Optimization, Springer, vol. 57(3), pages 677-693, November.
    9. J.-P. Penot & P. H. Quang, 1997. "Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 92(2), pages 343-356, February.
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