Continuous optimization

Given function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Find $\hat{x}$ such that:

$$f(\hat{x}) \leq \min_x f(x) + \epsilon$$ ^f11fcd

Have some function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Want to find $x^*$ such that: $$f(x^*) = \min_x f(x^*)$$

Or at least $\hat{x}$ which is close to a minimum e.g. #^f11fcd

Often have additional constraints such as $\mathbf{x} > 0$, $||\mathbf{x}||_2 \leq R$, $||\mathbf{x}||_1 \leq R$, $\mathbf{a}^T \mathbf{x} > c$

Also see: Convex Optimization notes (specific type of continuous optimization)