可微Quasiconvex function 本节考虑可微Quasiconvex function的性质 一阶条件 Suppose f:R^n \to R is differentiable. Then f is quasiconvex if and only if dom f is convex and \forall x,y\in dom f f(y)\leq f(x)\Rightarrow \nabla f(x)^T(y-x)\leq 0 回忆凸函数的一阶条件 f(y)\geq...
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\...
Concave and Convex Function What is Concave Function? Concave function is a function where the line segement between any two points of the function lies below or on the graph.[1] Mathematically, as fo...查看原文多载波MIMO信道下的波束成形设计 when the weights are in increasing order , and ...
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)≤...
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...
A 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) ...
Since a function is concave if and only if is convex, any result related to a convex function can easily be translated for a concave function. For this reason only the proofs related to convex functions are presented. For the sake of completeness, the corresponding results for the concave ...
Convex functions Page 3–3 Examples on Rn and Rm×n a?ne functions are convex and concave; all norms are convex examples on Rn ? ? examples on Rm×n (m × n matrices) ? a?ne function f (x ) = aT x + b ? p 1/p for p ≥ 1; ?x ? norms: ?x ?p = ( n ∞ = maxk...
A function that is convex will have a slope that is increasing. Additionally, all convex functions have only one minimum value. This is because the curve is opening upward, so the very tip of the bottom of the curve is the function's minimum. Concave curves on the other hand open up ...
Bottom: a convex function and it’s epigraph (which is a convex set). Perhaps not surprisingly (based on the above images), any continuous convex function is also a closed function. While the concept of a closed functions can technically be applied to both convex and concave functions, it ...