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.


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.


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

subject to


subject to

where denotes the dual cone of .

Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold.[1]

Semidefinite Program

The dual of a semidefinite program in inequality form

subject to

is given by

subject to


  • Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
  • MOSEK Software capable of solving conic optimization problems.
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