# Conic optimization

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.

## Definition

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

$f:C\to \mathbb {R}$ ## 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 $c^{T}x\$ subject to $Ax=b,x\in C\$ is

maximize $b^{T}y\$ subject to $A^{T}y+s=c,s\in C^{*}\$ ### Semidefinite Program

The dual of a semidefinite program in inequality form,

$x_{1}F_{1}+\cdots +x_{n}F_{n}+G\leq 0$ is given by

$\mathrm {tr} \ (F_{i}Z)+c_{i}=0,\quad i=1,\dots ,n$ $Z\geq 0$ 