Summary
In terms of n-th order Dini directional derivative with n positive integer we define n-pseudoconvex functions being a generalization of the usual pseudoconvex functions. Again with the n-th order Dini derivative we define n-stationary points, and prove that a point x 0 is a global minimizer of a n-pseudoconvex function f if and only if x 0 is a n-stationary point of f. Our main result is the following. A radially continuous function f defined on a radially open convex set in a real linear space is n-pseudoconvex if and only if f is quasiconvex function and any n-stationary point is a global minimizer. This statement generalizes the results of Crouzeix, Ferland, Math. Program. 23 (1982), 193–205, and Komlósi, Math. Program. 26 (1983), 232–237. We study also other aspects of the n-pseudoconvex functions, for instance their relations to variational inequalities.
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Ginchev, I., Ivanov, V.I. (2007). Higher-order Pseudoconvex Functions. In: Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol 583. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-37007-9_14
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DOI: https://doi.org/10.1007/978-3-540-37007-9_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37006-2
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