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

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
 
 MOSEK Software capable of solving conic optimization problems. Convex optimization