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Optimization theory: definition of convex function + generalized convex function

2022-07-04 02:14:00 【FakeOccupational】

Convex function

open Convex Set Ω ∈ R d , Letter Count u : Ω → R Open convex set \Omega\in R^d, function u:\Omega\rightarrow R open Convex Set Ω∈Rd, Letter Count u:Ω→R

u ( t x + ( 1 − t ) y ) ≤ u ( x ) + ( 1 − t ) u ( y ) . ∀ x , y , ∀ t ∈ [ x , y ] u(tx+(1-t)y)\leq u(x)+(1-t)u(y).\forall x,y ,\forall t \in[x,y] u(tx+(1−t)y)≤u(x)+(1−t)u(y).∀x,y,∀t∈[x,y]

Convex Letter Count One set yes One individual even To continue Letter Count A convex function must be a continuous function Convex Letter Count One set yes One individual even To continue Letter Count

Generalized convex function

open Set Ω ∈ R d , Letter Count u : Ω → R , Such as fruit u can By Tuo exhibition become R d On Of Convex Letter Count . The opening episode \Omega\in R^d, function u:\Omega\rightarrow R, If u It can be expanded into R^d The convex function over . open Set Ω∈Rd, Letter Count u:Ω→R, Such as fruit u can By Tuo exhibition become Rd On Of Convex Letter Count .
Than Such as ： to set Of Convex Letter Count u : Ω → R , stay repair Set R d \ Ω On Go to u by + ∞ such as ： Given convex function u:\Omega\rightarrow R, In complement R^d\backslash \Omega Up u by +\infty Than Such as ： to set Of Convex Letter Count u:Ω→R, stay repair Set Rd\Ω On Go to u by +∞

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