ExamplesA linear function is both convex and concave. f(x)=|x|f(x)=|x| is a convex function. f(x)=1xf(x)=1x is a convex function.TheoremLet f1,f2,...,fk:Rn→Rf1,f2,...,fk:Rn→R be convex functions. Consider the function f(x)=∑j=1kαjfj(x)f(x)=∑j=1kαjfj(x...
Log-determinant f(X) = \log \det X is concave on \bold{dom} \; f = S^n_{++} Proof for theses examples are written on P73.Sublevel sets The \alpha -sublevel set of a convex function f: \mathbb{R}^n \rightarrow \mathbb{R} is: ...
如果作为Rn+1的子集,集合epif是凸的,则我们将函数f定义为S上的凸函数。 4.1.3 定义(凹函数)concave function 空间\mathbb{R}^n上的子集S上的函数f是凹函数,当它的相反函数-f是凸函数时,这个函数在子集S上就是凹函数。 也即是,在空间\mathbb{R}^{n+1}中的子集epi(-f)是一个凸集时,函数f就是空间\...
Equivalently, a function : is concave if and only if hypo = () : ()is convex (Proposi- tion 3.7, Exercise 3.125) () (0) + (0) for every 0 (Exercise 4.67) : is at 0 if there exists a convex neighbor- hood of 0 such that for every 12 (1 + (1 )2) (1) + (1 )(2)...
The conjugate function Examples Quasiconvex functions Examples internal rate of return 内部收益率 Properties Log-concave and log-convex functions 正态分布的密度函数取 log 之后是一个常数减去一个凸的二次函数,因此是 concave 的 老师还讲了在任意集合上的均匀分布都是 log-concave 的,脑补一下,其概率密度...
f∈X;–α∈R,f∈Xthenαf∈X.Provethat:•afunctionisaffineiffitissimultaneouslyconcaveandconvex;("⇒"easy;"⇐"tricky:trywithf:R→R)•thesetofconcavefunctionsonasetDisaconeinthesetofallthefunctionsonD.(Isitalinearspace?)Regularityconditions:ContinuityTHEOREM4.Letf:D→Rbeaconcavefunction...
Perhaps not surprisingly (based on the above images), anycontinuous convex functionis also a closed function. While the concept of a closed functions can technically be applied to both convex andconcave functions, it is usually applied just to convex functions. Therefore, they are also calledclose...
Concave and Convex Function 技术标签:函数mathConcaveConvexmachine-learning 查看原文 多载波MIMO信道下的波束成形设计 whentheweights are in increasing order ,anditisthenaSchur-concavefunctiononeach carrier...isthenaSchur-concavefunction.Theproblem inconvexform (theobjectiveislinearandthe...
(Remark 4.14) Plotting the function (e.g. ) Apply properties to combinations of known functions (Examples 3.73 and 3.74) Hessian matrix (Proposition 4.1)is locally { convex concave } at () is { nonnegative nonpositive } definite is strictly locally { convex concave } at if () is { ...
Examples are automatically compiled from online sources to show current usage. Opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback. When the loss function has a convenient mathematical property called convexity, this procedure is guaranteed ...