Conic optimization

Conic optimization is a subfield of convex optimization that studies a class of structured convex optimization problems called conic optimization problems. A conic optimization problem consists of minimizing a convex function over the intersection of an affine subspace and a convex cone.

The class of conic optimization problems is a subclass of convex optimization problems and it includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming.

Definition

Given a real vector space X, a convex, real-valued function

f:C\to \mathbb {R}

defined on a convex cone $C\subset X$ , and an affine subspace ${\mathcal {H}}$ defined by a set of affine constraints $h_{i}(x)=0\$ , a conic optimization problem is to find the point $x$ in $C\cap {\mathcal {H}}$ for which the number $f(x)$ is smallest. Examples of $C$ include the positive semidefinite matrices $\mathbb {S} _{+}^{n}$ , the positive orthant $x\geq \mathbf {0}$ for $x\in \mathbb {R} ^{n}$ , and the second-order cone $\left\{(x,t)\in \mathbb {R} ^{n+1}:\lVert x\rVert \leq t\right\}$ . Often $f\$ is a linear function, in which case the conic optimization problem reduces to a semidefinite program, a linear program, and a second order cone program, respectively.