Second-order cone programming

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

${\displaystyle x\in {\mathcal {C}}_{t}}$

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

• Quadratic cone of dimension ${\displaystyle t}$ : ${\displaystyle {\mathcal {C}}_{t}=\left\{x\in \Re ^{n^{t}}:x_{1}\geq {\sqrt {\sum \limits _{j=2}^{n^{t}}x_{j}^{2}}}\right\}}$
• Rotated quadratic cone of dimension ${\displaystyle t}$ : ${\displaystyle {\mathcal {C}}_{t}=\left\{x\in \Re ^{n^{t}}:2x_{1}x_{2}\geq \sum \limits _{j=3}^{n^{t}}x_{j}^{2},~x_{1},x_{2}\geq 0\right\}}$

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