# Introduction

Outline

1. First-order algorithms
2. Decomposition and splitting
3. Second-order algorithms for unconstrained optimization
4. Interior point for conic optimization

### Convexity

- ∇^2(f (x) )> 0 is not necessary for strict convexity (cf., f (x) = x^4) but sufficient

- a differentiable function f is convex if and only if dom f is convex and (∇ f (x) − ∇ f (y)) ^T (x − y) ≥ 0 for all x, y ∈ dom f i.e., the gradient ∇ f : Rn → Rn is a monotone mapping

- Lipschitz Continuity 即利普希茨連續條件，是一個比通常連續更強的光滑性條件。 直覺上，利普希茨連續函數使用dual norm限制了函數改變的速度（從效果上感覺與限制二階導數ub類似），符合利普希茨條件的函數的斜率，必小於一個稱爲利普希茨常數的實數（該常數依函數而定）。

- ||∇ f (x) − ∇ f (y)||∗ ≤ L||x − y|| for all x, y ∈ dom f 則f爲L-smooth。此處的dual norm ,定義是對於所有normed vector v，線性關係zv的模的最大值。並且易見：

- Convexity vs Strict Convexity vs Strong Convexity (更加凸) (wiki)

- By using the property of convexity and Lipschitz continuity, upper and lower bound of f(x+delta) can be proved with a quadratic term. The equations reminds the second order Taylor approximation. After that, it can be proved by choosing step size smartly, gradient decent does the work at each iteration.