$$ \text{ (P) }\underset{x\in \R^n}{\min}f(x) \text{ s.t. } x\in C $$

• $f:\R^n\to\R$ convex and continuously differentiable
• $C\subseteq \R^n$ convex

If $\bar{x}$ is a local minimizer, then $\bar{x}$ is a global min

• level sets of convex functions

• first and second order optimization

  $\nabla f(x)=0$; $\nabla^2f(x)\ge0$

• Optimality for constrained problems

<aside> ➡️ Fact: If $f:\R^n\to\R$ convex, then the level sets $[f\le \tau]:=\{x\in \R^n \mid f(x)\in \tau\}$ are convex sets

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Suppose $\bar{x}\in C$ solves (P)

$\underbrace{f(\bar{x})}_{\equiv \tau}\le f(x), \forall x\in C$

$[f\le \tau]=\{x\mid f(x)\le \tau\}$

$[f\le \tau]\cap C$ ⇒ solution set

Take any $x,y\in [f\le \tau]$

Show that $\lambda x+(1-\lambda)y\in [f\le \tau]$ for $\lambda \in [0,1]$

$f(x)\le \tau$

$f(y)\le \tau$