Conic optimization

Conic optimization is a subfield of convex optimization. Given a real vector space X, a convex, real-valued function

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

defined on a convex cone $C\subset X$, and an affine subspace ${\mathcal {H}}$ defined by a set of affine constraints $h_{i}(x)=0\$, the problem is to find the point $x$ in $C\cap {\mathcal {H}}$ for which the number $f(x)$ is smallest. Examples of $C$ include the positive semidefinite matrices $\mathbb {S} _{+}^{n}$, the positive orthant $x\geq \mathbf {0}$ for $x\in \mathbb {R} ^{n}$, and the second-order cone $\left\{(x,t)\in \mathbb {R} ^{n+1}:\lVert x\rVert \leq t\right\}$. Often $f\$ is a linear function, in which case the conic optimization problem reduces to a semidefinite program, a linear 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 $c^{T}x\$

subject to $Ax=b,x\in C\$

is

maximize $b^{T}y\$

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

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

Semidefinite Program

The dual of a semidefinite program in inequality form,

minimize $c^{T}x\$ subject to

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

is given by

maximize $\mathrm {tr} \ (GZ)\$ subject to

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

$Z\geq 0$

Refrences

• Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo, 2005.

External links

• Stephen Boyd and Lieven Vandenberghe, Convex Optimization (book in pdf)
• MOSEK Software capable of solving conic optimization problems.