Some properties of E -convex functions

Recently, it was shown by Youness [E.A. Youness, On E -convex sets, E -convex functions and E -convex programming, Journal of Optimization Theory and Applications, 102 (1999) 439–450] that many results for convex sets and convex functions actually hold for a wider class of sets and functions, called...

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Bibliographic Details
Published inApplied mathematics letters Vol. 18; no. 9; pp. 1074 - 1080
Main Authors Syau, Yu-Ru, Stanley Lee, E.
Format Journal Article
LanguageEnglish
Published Oxford Elsevier Ltd 01.09.2005
Elsevier
Subjects
[formula omitted]-convexity
Calculus of variations and optimal control
Convexity
Exact sciences and technology
Generalized convexity
Mathematical analysis
Mathematics
Measure and integration
Nonlinear programming
Sciences and techniques of general use
Convexity
Nonlinear programming
Generalized convexity
E -convexity
Convex set
Level set
Optimization method
E-convexity
Non linear programming
Convex programming
E convexity
Applied mathematics
Convex function
Programming theory
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Summary:Recently, it was shown by Youness [E.A. Youness, On E -convex sets, E -convex functions and E -convex programming, Journal of Optimization Theory and Applications, 102 (1999) 439–450] that many results for convex sets and convex functions actually hold for a wider class of sets and functions, called E -convex sets and E -convex functions. We introduce the concept of E -quasiconvex functions and strictly E -quasiconvex functions, and develop some basic properties of E -convex and E -quasiconvex functions. For a real-valued function f defined on a nonempty E -convex set M , we show under the convexity condition of E ( M ) , that f is E -quasiconvex (resp. strictly E -quasiconvex) if and only if its restriction to E ( M ) is quasiconvex (resp. strictly quasiconvex). Similarly, we show under the convexity condition of E ( M ) , that f is E -convex (resp. strictly E -convex) if and only if its restriction to E ( M ) is convex (resp. strictly convex). In addition, under the convexity condition of E ( M ) , a characterization of an E -quasiconvex function in terms of the lower level sets of its restriction to E ( M ) is also given. Finally, examples in nonlinear programming problem are used to illustrate the applications of our results.
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2004.09.018