Notes on convex optimization

Identify convex constraints

Inequality constraints

• If $f$ is convex, the constraint $f(x) \leq a \in \mathbb{R}$ is convex.

Take $x_1,x_2$ that satisfy that constraint and $\alpha \in [0,1]$, àthen if $f$ is convex, we have \(f(\alpha x_1 + (1-\alpha) x_2) \leq \alpha f(x_1) + (1-\alpha) f(x_2) \leq a\).