Results 331 to 340 of about 10,934,899 (375)

Convex Sets and Convex Functions

2002
This chapter explores sets that can be represented as intersections of (a possibly infinite number of) halfspaces of Rn . 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 natural generalization of linear functions.
Ulrich Faigle, Walter Kern, Georg Still
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Second-Order Lower Bounds on the Expectation of a Convex Function

Mathematics of Operations Research, 2005
We develop a class of lower bounds on the expectation of a convex function. The bounds utilize the first two moments of the underlying random variable, whose support is contained in a bounded interval or hyperrectangle.
S. Dokov, D. Morton
semanticscholar   +1 more source

Convex Sets and Convex Functions

2014
Convex sets and functions have been studied since the nineteenth century; the twentieth century literature on convexity began with Bonnesen and Fenchel’s book [1], subsequently reprinted as [2].
Dan A. Simovici, Chabane Djeraba
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Convex Sets and Convex Functions [PDF]

, 1979
Because of their useful properties, the notions of convex sets and convex functions find many uses in the various areas of Applied Mathematics. We begin with the basic definition of a convex set in n-dimensional Euclidean Space (En), where points are ordered n-tuples of real numbers such as x’ = (x1, x2,…, xn) and y’ = (y1, y2,…,yn).
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Some properties of the Hessian Matrix of a Strictly Convex Function.

, 1962
A strictly convex function of n real variables is any continuously differentiable funetion with the property (1) below. Strictly convex functions arise naturally in a number of physical theories, notably, in thermodynamics and elasticity theory.
R. Toupin, B. Bernstein
semanticscholar   +1 more source

Convex Sets and Convex Functions

2016
The first chapter introduces the fundamental concepts and conclusions of functional analysis so that readers can have a foundation for going on reading this book successfully and can also understand notations used in the book. The arrangement of this chapter is as follows: The first section deals with normed linear spaces and inner product spaces which
Zhengzhi Han, Xiushan Cai, Jun Huang
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The smallest convex extensions of a convex function

, 1992
Let C be a solid convex subset of a linear space X and be an algebraically-1.s.c. convex function. We prove the existence of a smallest convex extension of f to the whole space X, i.e.
F. Dragomirescu, C. Ivan
semanticscholar   +1 more source

On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function

Computational optimization and applications, 2016
Xingju Cai, Deren Han, Xiaoming Yuan
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On (?,a)-convex functions

Archiv der Mathematik, 1995
Matkowski, Janusz, Pycia, Marek
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Semi-Infinite Optimization under Convex Function Perturbations: Lipschitz Stability

Journal of Optimization Theory and Applications, 2011
Nguyen Quang Huy, Jen-Chih Yao
semanticscholar   +1 more source