Second-order cone programming

A Second order cone program (SOCP) is a convex optimization problem of the form

minimize ${\displaystyle f^{T}x}$ subject to

${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}}$, ${\displaystyle i\in \{1,\dots ,m\}}$

${\displaystyle Fx=g}$

where ${\displaystyle x}$ is the optimization variable, ${\displaystyle A_{i}\in \mathbb {R} ^{{n_{i}}\times n}}$, and ${\displaystyle F\in \mathbb {R} ^{p\times n}}$. When ${\displaystyle A_{i}=0}$ for ${\displaystyle i\in \{1,\dots ,m\}}$, the SOCP reduces to a linear program. When ${\displaystyle c_{i}=0}$ for ${\displaystyle i\in \{1,\dots ,m\}}$, the SOCP is equivalent to a Quadratically constrained quadratic program.