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
minimize subject to
is
maximize subject to
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
minimize subject to
is given by
maximize subject to
References
External links
Splitting Conic Solver (SCS) Package for solving large-scale convex cone problems. MOSEK Software capable of solving conic optimization problems.