Lec03 Convex Functions

1. Basic properties and examples

1.1 Definitiion

convex funtion

dom f is a convex set and

$$ f(θx + (1 - θ)y) ≤ θf(x) + (1 - θ)f(y) $$ for all \(x,y\in dom\ f ,0\leq \theta \leq 1\)

strictly convex funtion

dom f is a convex set and $$ f(θx + (1 - θ)y) ≤ θf(x) + (1 - θ)f(y) $$ for all \(x\neq y\in dom\ f ,0<\theta < 1\)

concave

凹

Examples on R

• 

Examples on \(R^n\) and \(R^{m\times n}\)

• 

一个判断法则

• 判断 \(f:R^n ->R\) 是否是凸的
• 只需判断函数 \(g:R->R\ g(t)=f(x+tv),\ dom g =\{ t| x+tv \in dom \ f \}\)是否是凸的。
• 举一个例子
• 

1.2 Extended-value extension

we define \(\tilde{f}\) is the extension of f,we get

\[ \tilde{f}(x)= \begin{cases} f(x)\quad x \ in\ dom \ f \\ \infty \quad \ \ \ x \notin \ dom \ f \end{cases} \]

so we can extend the funtion to \(R^n\)

1.3 First-order condition

1.4 Second-order conditions

1.5 Epigraph and sublevel set

上境图和\(\alpha\)-下水平集

1.6 Jensen's inequality

basic inequality

if f is convex,then for \(0\leq \theta \leq 1\) $$ f(θx + (1 - θ)y) ≤ θf(x) + (1 - θ)f(y) $$

extension

if f is convex then, \(x_1,...,x_k \in dom \ f ,\theta_1,...,\theta_k\geq 0\),\(\theta_1+...+\theta_k=1\)，则 $$ f(\theta_1x_1+···+\theta_kx_k)\leq \theta_1f(x_1)+...+\theta_kf(x_k) $$ 同理可以应用于积分 用概率来衡量，我们可以得到 $$ f(Ex)\leq Ef(x) $$

Holder不等式

• Contents

---

2. Operations that preserve convexity

1. 非负加权求和（积分）
2. 复合仿射映射 \(f(Ax+b)\) is convex if f is convex
3. 逐点最大与逐点上确界
4. 标量复合
   
5. 矢量复合
   
6. 最小化
7. 透视函数

3.The conjugate function 共轭函数

conjugate function

\[f^*(y)=sup_{x\in dom \ f}(y^Tx-f(x))\]

不论f是否是凸函数，\(f^*\)均为凸函数。

---

4.Quasiconvex functions 拟凸函数

definition

如果所有的下 \(\alpha\) 水平集

\[S_\alpha= \{x\in dom \ f | f(x)\leq \alpha\}\]

都为凸集，那么为拟凸函数

properties

1. modified Jensen inequality: for quasiconvex f $$ 0 ≤ θ ≤ 1 =⇒ f(θx + (1 - θ)y) ≤ max{f(x), f(y)} $$
2. first-order condition: differentiable f with cvx domain is quasiconvex iff $$ f(y) ≤ f(x) =⇒ ∇f(x)^T (y - x) ≤ 0 $$

---

5.Log-concave and log-convex functions

log-concave

a positive function f is log-concave if log f is concave:

\[f(θx + (1 - θ)y) ≥ f(x)^θf(y)^{1-θ}\ for\ 0 ≤ θ ≤ 1\]

log-convex

f is log-convex if log f is convex

properties

---

6.Convexity with respect to generalized inequalities