On strongly E-convex sets and strongly E-convex cone sets

Abstract

-convex sets and -convex functions, which are considered as an important class of generalized convex sets and convex functions, have been introduced and studied by Youness [5] and other researchers. This class has recently extended, by Youness, to strongly -convex sets and strongly -convex functions. In these generalized classes, the definitions of the classical convex sets and convex functions are relaxed and introduced with respect to a mapping . In this paper, new properties of strongly -convex sets are presented. We define strongly -convex hull, strongly -convex cone, and strongly -convex cone hull and we proof some of their properties. Some examples to illustrate the aforementioned concepts and to clarify the relationships between them are established.

Computation of the Hausdorff Distance between Two Compact Convex Sets

“…In particular, is strongly convex when . Furthermore, the Cartesian product of two strongly convex sets is strongly convex [ 35 ]. The homotopy method succeeds because the early sets are more rounded and the objective generates fewer local maxima.…”

Strongly (E,F)-convexity with applications to optimization problems

“…11 Let 𝑀 = ℝ, 𝑓: ℝ → ℝ, and 𝐸, 𝐹: ℝ → ℝ such that 𝑚 1 , 𝑚 2 ∈ ℝ and 𝜆 ∈ [0,1]. First, we show that 𝑓(𝜆(𝐸𝑚 1 ) + (1 − 𝜆)(𝐹𝑚 2 )) ≤ 𝜆𝑓(𝐸𝑚 1 ) + (1 − 𝜆)𝑓(𝐹𝑚 2 ).…”

“…For more results on 𝐸-convexity see e.g., [17,15,10,1]. Youness and Emam [21] extended the class of 𝐸-convex sets and 𝐸convex functions into strongly 𝐸-convex sets and 𝐸-convex functions (see Definitions 1.3 and 1.8), respectively, and studied their properties (for more recent paper on strongly 𝐸-convex sets and strongly 𝐸-convex cone sets, see [11]). The class of semi 𝐸-convex functions is extended into the class of strongly semi 𝐸-conex functions by Youness and Emam [22].…”