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

$f:C \to \mathbb R$

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 \$, a conic optimization 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 orthant $\mathbb{R}_+^n = \left\{ x \in \mathbb{R}^n : \, x \geq \mathbf{0}\right\}$, positive semidefinite matrices $\mathbb{S}^n_{+}$, and the second-order cone $\left \{ (x,t) \in \mathbb{R}^{n}\times \mathbb{R} : \lVert x \rVert \leq t \right \}$. 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 $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^* \$

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

$Z \geq0$

References

1. "Duality in Conic Programming" (PDF).

External links

• 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.