Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone. The class of conic optimization problems 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 defined on a convex cone C \subset X, and an affine subspace \mathcal{H} defined by a set of affine constraints, a conic optimization problem is to find the point x in for which the number f(x) is smallest. Examples of C include the positive orthant, positive semidefinite matrices , and the second-order cone. Often f \ is a linear function, in which case the conic optimization problem reduces to a linear program, a semidefinite program, and a second order cone program, respectively.

Duality

Certain special cases of conic optimization problems have notable closed-form expressions of their dual problems.

Conic LP

The dual of the conic linear program is where C^* denotes the dual cone of C . Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold.

Semidefinite Program

The dual of a semidefinite program in inequality form is given by