Conic optimization is a subfield of convex optimization that studies a class of structured convex optimization problems called conic optimization problems. A conic optimization problem consists of minimizing a convex function over the intersection of an affine subspace and a convex cone.
The class of conic optimization problems is a subclass of convex optimization problems and it 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 , 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 semidefinite matrices , the positive orthant for , and the second-order cone . Often is a linear function, in which case the conic optimization problem reduces to a semidefinite program, a linear 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
where denotes the dual cone of .
The dual of a semidefinite program in inequality form,
is given by
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