D 包含於 R^n is a convex set
f: D → R is concave in D
We extend the domain f to a bigger set D* 包含 D
by f*(x) = { f(x) for x in D
{
{ f(x*) - k |x-x*| for x in D*\D , k > 0
where x* = arg min |x-y|
y in D
( 最靠近 D 的點 , unique by convex of D)
也就是說 把f在D 的邊界用平面直接往下拉，速度是 k
Prove that f* is quasi-concave in D* for k large enough
quasi-concave :
f( λx + (1-λ)y ) ≧ min {f(x) , f(y)} for all 0 ≦ λ ≦ 1
or equivalently, the set {x: f(x) ≧ c} is convex for all c
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◆ From: 140.113.211.193
※ 編輯: GSXSP 來自: 140.113.211.193 (10/02 21:19)
※ 編輯: GSXSP 來自: 140.113.211.193 (10/03 18:35)