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

: $f:C \to \mathbb R$

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 \$

: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.

=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

{{Reflist|refs=

{{cite web|title=Duality in Conic Programming|url=https://people.smp.uq.edu.au/YoniNazarathy/teaching_projects/studentWork/Duality.pdf}}

}}

External links

{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |accessdate=October 15, 2011}}
[http://www.mosek.com MOSEK] Software capable of solving conic optimization problems.

Category:Convex optimization