Convex functions

1 凸优化

1.1 定义

A function f:Rn→R is convex if domf is a convex set and if for all x,y∈domf, and θ with 0≤θ≤1, we have

f(θx+(1−θ)y)≤θf(x)+(1−θ)f(y).

1.2 一阶条件

Suppose f is differentiable. Then f is convex if and only if domf is convex and

f(y)≥f(x)+▽T(y−x)

holds for all x,y∈domf.

1.3 二阶条件

We now assume that f is twice differentiable, that is, its Hessian or second order derivative ▽2f exists at each point in domf, which is open. Then f is convex if and only if domf is convex and its Hessian is positive semidefinite: for all x∈domf,

▽2f(x)⪰0.

1.4 eipgraph

The epigraph of a function f:Rn→R is defined as

epif={(x,t)|x∈domf,f(x)≤t}.

The link between convex sets and convex functions is via the epigraph: A function is convex if and only if its epigraph is a convex set.

2 Operations that preserve convexity

2.1 Nonnegative weighted sums

f=w1f1+⋯+wmfm

2.2 Composition with an affine mapping

g(x)=f(Ax+b)

2.3 Pointwise maximun and supremum

f(x)=max{f1(x),f2(x)}

2.4 Composition？

2.5 Minimization?

2.6 Perspective of a function?

3 The conjugate function

Let f:Rn→R. The function f∗:Rn→R, defined as

f∗(y)=supx∈domf(yTx−f(x)),

is called the conjugate of the function f.

We see immediately that f∗ is a convex function, since it is the pointwise supremum of a family of convex (indeed, affine) function of y. This is true whether or not f is convex.