Conic optimization

Short description: Subfield of convex optimization

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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.

## Contents

* 1 Definition
* 2 Duality
* 2.1 Conic LP
* 2.2 Semidefinite Program
* 3 References
* 4 External links

## Definition

Given a real vector space X, a convex, real-valued function

f:C->ℝ

defined on a convex cone C⊂X, and an affine subspace ℋ defined by a set of affine constraints hi(x)=0 , a conic optimization problem is to find the point x in C∩ℋ for which the number f(x) is smallest.

Examples of C include the positive orthant ℝ+n={x∈ℝn:x≥𝟎}, positive semidefinite matrices 𝕊+n, and the second-order cone {(x,t)∈ℝn×ℝ:‖x‖≤t}. Often f 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 cTx
subject to Ax=b,x∈C

maximize bTy
subject to ATy+s=c,s∈C*

where C* denotes the dual cone of C .

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

minimize cTx
subject to x1F1+⋯+xnFn+G≤0

is given by

maximize tr (GZ)
subject to tr (FiZ)+ci=0,i=1,…,n
Z≥0

## References

1. ↑ "Duality in Conic Programming". https://people.smp.uq.edu.au/YoniNazarathy/teaching_projects/studentWork/Duality.pdf.

## External links

* Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 978-0-521-83378-3. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf. Retrieved October 15, 2011.
* MOSEK Software capable of solving conic optimization problems.

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