Convex function

2016年3月1日 — Lemma 1. Strong convexity ⇒ Strict convexity ⇒ Convexity. (But the converse of neither implication is true.) Proof: The fact that strict ...

Proposition: A function f : Rn → R is convex if and only if epif ⊂ Rn+1 is convex. Proof. Assume f is convex. Let (x1,t1), (x2,t2) ∈ epif and λ ∈ [0, 1] ...

2021年9月5日 — Theorem 4.6.1 ... Let I be an interval of R. A function f:I→R is convex if and only if for every λi≥0,i=1,…,n, with ∑ni=1λi=1 (n≥2) and for ...

In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function ...

2.1 That's great, but how do I prove that a function is convex? 1. If you know calculus, take the second derivative. It is a well-known fact that if the second ...

To prove convexity, you need an argument that allows for all possible values of x1, x2, and λ, whereas to disprove it you only need to give one set of values.

2019年8月16日 — What you gave is the standard definition of a convex function. If f is supposed to be continuous, it is enough to check that.

2022年8月2日 — Consider that f(x)=x2 has f″(x)=2>0, so f is a convex function. It is also strongly convex too (and hence strictly convex) with strong convexity ...