Conic optimization

From Wikipedia, the free encyclopedia

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.

Contents

1 Duality
- 1.1 Conic LP
- 1.2 Semidefinite Program
2 External links

Certain special cases of conic optimization problems have notable closed-form expressions of their dual problems.

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 \$ .

The dual of a semidefinite program in inequality form,

minimize $c^T x \$ subject to

$x_1 F_1 + \cdots + x_n F_n + G \leq 0$

$Ax = b \$

is given by

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

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

$Z \geq0$

Stephen Boyd and Lieven Vandenberghe, Convex Optimization (book in pdf)

Retrieved from "http://en.wikipedia.org../../../c/o/n/Conic_optimization.html"

Category: Optimization

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