Conic optimization
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, and an affine subspace defined by a set of affine constraints, a conic optimization problem is to find the point in for which the number is smallest.
Examples of include the positive orthant, positive semidefinite matrices, and the second-order cone. Often 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 denotes the dual cone of.
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
