Second order cone programming

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Second order cone programing (SOCP) can be seen as a generalization of linear programming and quadratic programming. Simply put, a conic optimization problem is a linear optimization problem plus a number of constraints of the form

x \in \mathcal{C}_t

where \mathcal{C}_t is a second order cone of dimension t. Two types of second order cones are:

  • Quadratic cone of dimension t : \mathcal{C}_t = \left \{ x \in \real^{n^t}: x_1 \geq \sqrt{\sum\limits_{j=2}^{n^t} x_j^2} \right \}
  • Rotated quadratic cone of dimension t : \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 \}

Several software packages exists for solving SOCP problems, such as MOSEK and SeDuMi.

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