Convex functions

Recall that a function \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) is strictly convex if

\(\hspace{2cm}∀x,y,x≠y,∀λ∈(0,1),\)\(f(λx+(1-λ))<λf(x)+(1-λ)f(y).\)

Like we mentioned before, if \(f\) is strictly convex, then \(f\) is convex (this is obvious from the definition) but the converse is not true, take for example \(f(x)=x,x∈\mathbb{R}^{n}.\)

\(\hspace{5cm}∇^{2}f(x)>0,∀x∈Ω⇒f\) strictly convex on \(Ω\).

The converse is not true through.

A function \(f\) is strictly convex on \(Ω⊆\mathbb{R}^{n}\) if and only if

\(\hspace{5cm}f(y)>f(x)+∇f^{T}(x)(y-x),∀x,y∈Ω,x≠y.\)

• There are similar characterizations for strong convex functions. For example, \(f\) is strongly convex if and only if there exists \(m>0\) such that

  \(\hspace{1cm}f(y)≥f(x)+∇^{2}f(x)(y-x)+m‖y-x‖^{2},∀x,y∈dom(f),\)

or if and only if there exists \(m>0\) such that \(∇^{2}f(x)≥mI,∀x∈dom(f).\)

• One of the main uses of strict convexity is to ensure uniqueness of the optimal solution.