其中\({\sigma _1}(X) \ge \ldots \ge \sigma (X) \ge 0\)为矩阵X的从大到小排序的奇异值。 指示函数(Indicator function): 如果C为凸,那么其指示函数为:\({I_C}(x) = \left\{ \begin{array}{l}0,x \in C\\\infty ,x \notin C\end{array} \right.\)为凸函数。
Convex Set and Convex Function凸集与凸函数 Welcome To My Blog Rockafeller说:”优化问题的分水岭不是线性和非线性,而是凸性和非凸性” 两点连线上的点 在介绍凸集和凸函数之前,先来看一个与之有关的基本问题: 如下图,已知空间中有B,C两点,在给定两点坐标的情况下如何量化B,C连线上的任意一...
第3讲 凸集、凸函数、凸规划 凸性(Convexity)是最优化理论必须涉及到基本概念.具有凸性的非线性规划模型是一类特殊的重要模型，它在最优化的理论证明及算法研究中具有非常重要的作用. 凸集(ConvexSet)凸函数(ConvexFunction)凸规划(ConvexProgramming) 线性组合(linearCombination)mnixi,其中iR...
AAE 590 : Introduction to Convex Optimization Convex set and function Convex setSun, Dengfeng
if dom f is a convex set, and 0≤θ≤1, all x,y∈domf: f(θx+(1−θ)y)≤θf(x)+(1−θ)f(y). 则函数f是凸函数(convex function) 如果把定义增强,则是严格凸(strictly convex)。if 0<θ<1 and x≠y, and f(θx+(1−θ)y)<θf(x)+(1−θ)f(y). Note: 如果−...
The link between convex sets and convex functions is via the epigraph. A function is convex iff. its epigraph is a convex set. Similarly, a function f is concave iff. its hypograph defined as: \bold{hypo}\; f = \{(x, t) | t \le f(x)\} is a convex set....
Definition 2: Convex function (Boyd & Vandenberghe, 2004; Du et al., 2008). if function f(x) is a convex, set C is a convex set, for all x,y∈C, 0≤ θ≤1, then (2.4)f(θx+(1-θ)y)⩽θf(x)+(1-θ)f(y) Definition 3 Convex optimization problem (Boyd & Vandenberghe,...
Let us observe that the convexity of S is also necessary since S is the set of minima of dS and it is an exercise to prove that the set of minima of a convex function is convex. Another property of the distance function when S is a closed convex set is that the inf in its definit...
Convexandconcavefunctions 2.5.Convex and concave functions Let f be a real-valued function f:S→R.In what follows,we assume that S⊆R n is a convex set.The function f is convex on a set S if∀x1,x2∈S,∀λ∈[0,1]such thatλx1+(1−λ)x2∈S,f(λx1+(1−λ)x2)≤...
1. Affine and convex sets 1.1 Affine sets A set \(C\in\R^n\) is affine if the line through any two distinct points in \(C\) lies in \(C\), which mea