This chapter explores sets that can be represented as intersections of (a possibly infinite number of) halfspaces of R n . As will be shown, these are exactly the closed convex subsets. Furthermore, convex functions are studied, which are closely connected to convex sets and provide a ...
像perspective function一样, linear-fractional functions也维护了凸性。如果C是凸的并且位于f的定义域中( i.e., c^Tx + d \gt 0 对于 x \in C) ,那么它的图像f(C)就是凸的。同样的,如果 C\subseteq R^m 那么f^{-1}(x) 的反图像时也是凸的。
(2001). Separation Theorems for Convex Sets and Convex Functions with Invariance Properties. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, ...
Quasiconvex functionsDefinitionA function \mathcal R^n\to \mathcal R is quasiconvex if its domain and all sublevel sets S_\alpha =\{x\in dom f|f(x)\leq \alpha \},\forall \alpha \in \mathcal Rare con…
For each of the four families of convex sets: the Euclidean balls, the cubes, the regular cross-polytopes and the regular symplexes of dimensions (n), the limiting entire functions, as (n) tends to infinity, are calculated explicitly.Katsnelson, Victor...
Suitable for advanced undergraduates and graduate students, this text introduces the broad scope of convexity by highlighting diverse applications. Topics include characterizations of convex sets, polytopes, duality, optimization, and convex functions. Exercises appear throughout the text, with solutions, hi...
The aim of this paper is to present a geometric characterization of even convexity in separable Banach spaces, which is not expressed in terms of dual functionals or separation theorems. As an application, an analytic equivalent definition for the class of evenly quasiconvex functions is derived.关...
strong quasiconvex functionsnon empty interior/ C1160 Combinatorial mathematicsIt is shown that the level sets are bounded for any xC if the function f:C R is strong quasiconvex and C R n is a convex set with nonempty interior. Thus we have generalized one of the results from Vial [3]....
Convex Functions 2.1 Convex Sets and Convex Functions A C in a Hilbert space X is a set with the following property: for every x,y∈C,C contains the segment [x,y]={tx+(1-t)y:t∈[0,1]} (see for instance, Fig. 2.1). The next proposition summarizes some elementary properties of ...
Chapter 2:Convex Sets(凸集) Chapter 3:Convex functions(凸函数) Chapter 4:Convex optimization problems Chapter 5: Lagrangian duality (拉格朗日对偶) Part II: Applications(主要介绍凸优化是如何应用在实际中的) Part III: Algorithms unconstrained optimization ...