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A '''Second order cone programingprogram''' ('''SOCP''') can be seen asis a generalization of [[linearconvex programmingoptimization]] andproblem [[quadraticof programming]].the Simply put, a conic optimization problem is aform
linear optimization problem plus a number of constraints of the form
minimize <math>f^T x </math> subject to
:<math>
x \in \mathcal{C}_t
<math>\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i</math>, <math>i\in\{1,\dots,m\}</math>
</math>
where <math>\mathcal{C}_t</math> is a second order cone of dimension <math> t </math>. Two types of second order cones are:
<math>Fx = g</math>
* Quadratic cone of dimension <math> t </math> : <math> \mathcal{C}_t = \left \{ x \in \real^{n^t}: x_1 \geq \sqrt{\sum\limits_{j=2}^{n^t} x_j^2} \right \} </math>
* Rotated quadratic cone of dimension <math> t </math> : <math> \mathcal{C}_t = \left \{ x \in \real^{n^t}: 2 x_1 x_2 \geq \sum\limits_{j=3}^{n^t} x_j^2,~ x_1,x_2 \geq 0 \right \} </math>
where <math>x</math> is the optimization variable, <math>A_i \in \mathbb{R}^{{n_i}\times n}</math>, and <math>F \in \mathbb{R}^{p\times n}</math>. When <math>A_i = 0</math> for <math>i\in\{1,\dots,m\}</math>, the SOCP reduces to a [[linear program]]. When <math>c_i = 0 </math> for <math>i\in\{1,\dots,m\}</math>, the SOCP is equivalent to a [[Quadratically constrained quadratic program]].
Several software packages exists for solving SOCP problems, such as [http://www.mosek.com MOSEK] and [[SeDuMi]].
[[Category:Optimization]]
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