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 defined by a set of affine, a conic optimization problem is smallest.

include the positive orthant, positive semidefinite, and the second-order cone 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

denotes the dual cone.

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

References

External links

• Splitting Conic Solver (SCS) Package for solving large-scale convex cone problems.

• MOSEK Software capable of solving conic optimization problems.