Convex functions

1 凸优化

1.1 定义

A function $f:R^n\to R$ is convex if $domf$ is a convex set and if for all $x,y\in domf$, and $\theta$ with $0\le \theta \le 1,$ we have
$$f(\theta x+(1-\theta )y)\le \theta f(x)+(1-\theta )f(y).$$

1.2 一阶条件

Suppose $f$ is differentiable. Then $f$ is convex if and only if $domf$ is convex and
$$f(y)\ge f(x)+{▽}^{T}f(x)(y-x)$$ holds for all $x,y\in domf$.

1.3 二阶条件

We now assume that $f$ is twice differentiable, that is, its Hessian or second order derivative ${▽}^{2}f$ 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 $x\in domf$, $${▽}^{2}f(x)⪰0.$$

1.4 eipgraph

The epigraph of a function $f:R^n\to R$ is defined as $$epif=\{(x,t)|x\in domf,f(x)\le t\}.$$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

$$f=w_1{f}_1+\cdots +w_m{f}_m$$

2.2 Composition with an affine mapping

$$g(x)=f(Ax+b)$$

2.3 Pointwise maximun and supremum

$$f(x)=max\{f_1(x),f_2(x)\}$$

2.4 Composition？

2.5 Minimization?

2.6 Perspective of a function?

3 The conjugate function

Let $f:R^n\to R.$ The function $f^{\ast }:R^n\to R,$ defined as
$$f^{\ast }(y)=\underset{x\in domf}{\sup }(y^{T}x-f(x)),$$
is called the conjugate of the function $f$.

We see immediately that $f^{\ast }$ 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.