Constrained convex optimization

Solve a convex minimization problem with additional convex constraints, $$\min_{x \in \mathcal{S}}f(\mathbf{x})$$ where $\mathcal{S}$ is a convex set.

Examples:

• norm constraint: minimize $||\mathbf{Ax} − \mathbf{b}||_2$ subject to $||\mathbf{x}||_2 ≤ λ$.
• positivity constraint: minimize $f(\mathbf{x})$ subject to $\mathbf{x} \geq 0$
• linear constraint: minimize $\mathbf{c}^\intercal \mathbf{x}$ subject to $\mathbf{Ax} \leq \mathbf{b}$