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

${\displaystyle f:C\to \mathbb {R} }$

defined on a convex cone ${\displaystyle C\subset X}$, and an affine subspace ${\displaystyle {\mathcal {H}}}$ defined by a set of affine constraints ${\displaystyle h_{i}(x)=0\ }$, a conic optimization problem is to find the point ${\displaystyle x}$ in ${\displaystyle C\cap {\mathcal {H}}}$ for which the number ${\displaystyle f(x)}$ is smallest.

Examples of ${\displaystyle C}$ include the positive orthant ${\displaystyle \mathbb {R} _{+}^{n}=\left\{x\in \mathbb {R} ^{n}:\,x\geq \mathbf {0} \right\}}$, positive semidefinite matrices ${\displaystyle \mathbb {S} _{+}^{n}}$, and the second-order cone ${\displaystyle \left\{(x,t)\in \mathbb {R} ^{n}\times \mathbb {R}$ :\lVert x\rVert \leq t\right\}} . Often ${\displaystyle f\ }$ 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 ${\displaystyle c^{T}x\ }$

subject to ${\displaystyle Ax=b,x\in C\ }$

is

maximize ${\displaystyle b^{T}y\ }$

subject to ${\displaystyle A^{T}y+s=c,s\in C^{*}\ }$

where ${\displaystyle C^{*}}$ denotes the dual cone of ${\displaystyle C\ }$.

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 ${\displaystyle c^{T}x\ }$

subject to ${\displaystyle x_{1}F_{1}+\cdots +x_{n}F_{n}+G\leq 0}$

is given by

maximize ${\displaystyle \mathrm {tr} \ (GZ)\ }$

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

${\displaystyle Z\geq 0}$