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.
Given a real vector space X, a convex, real-valued function defined on a convex cone
defined by a set of affine constraints
, a conic optimization problem is to find the point
include the positive orthant
, positive semidefinite matrices
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.
Certain special cases of conic optimization problems have notable closed-form expressions of their dual problems.
The dual of the conic linear program is where
Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold.
[1] The dual of a semidefinite program in inequality form is given by