Solve a convex minimization problem with additional convex constraints,

minx∈𝒮⁡f(𝐱)\min_{x \in \mathcal{S}}f(\mathbf{x})

where 𝒮\mathcal{S} is a convex set.

Examples:

  • norm constraint: minimize ||𝐀𝐱−𝐛||2||\mathbf{Ax} − \mathbf{b}||_2 subject to ||𝐱||2≤λ||\mathbf{x}||_2 ≤ λ.
  • positivity constraint: minimize f(𝐱)f(\mathbf{x}) subject to 𝐱≥0\mathbf{x} \geq 0
  • linear constraint: minimize 𝐜⊺𝐱\mathbf{c}^\intercal \mathbf{x} subject to 𝐀𝐱≤𝐛\mathbf{Ax} \leq \mathbf{b}