# Second-order cone programming

(Redirected from 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
${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots ,m}$
${\displaystyle Fx=g\ }$

where the problem parameters are ${\displaystyle f\in \mathbb {R} ^{n},\ A_{i}\in \mathbb {R} ^{{n_{i}}\times n},\ b_{i}\in \mathbb {R} ^{n_{i}},\ c_{i}\in \mathbb {R} ^{n},\ d_{i}\in \mathbb {R} ,\ F\in \mathbb {R} ^{p\times n}}$, and ${\displaystyle g\in \mathbb {R} ^{p}}$. (${\displaystyle \lVert x\rVert _{2}}$ is the Euclidean norm and (${\displaystyle T}$ indicates Transpose) Here ${\displaystyle x\in \mathbb {R} ^{n}}$ is the optimization variable. [1] When ${\displaystyle A_{i}=0}$ for ${\displaystyle i=1,\dots ,m}$, the SOCP reduces to a linear program. When ${\displaystyle c_{i}=0}$ for ${\displaystyle i=1,\dots ,m}$, the SOCP is equivalent to a convex quadratically constrained linear program.

Convex Quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint. Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semi definite program.

SOCPs can be solved with great efficiency by interior point methods.

Consider a quadratic constraint of the form

${\displaystyle x^{T}A^{T}Ax+b^{T}x+c\leq 0.}$

This is equivalent to the SOC constraint

${\displaystyle \left\|{\begin{matrix}(1+b^{T}x+c)/2\\Ax\end{matrix}}\right\|_{2}\leq (1-b^{T}x-c)/2.}$

## Example: Stochastic linear programming

Consider a stochastic linear program in inequality form

minimize ${\displaystyle \ c^{T}x\ }$
subject to
${\displaystyle P(a_{i}^{T}x\leq b_{i})\geq p,\quad i=1,\dots ,m}$

where the parameters ${\displaystyle a_{i}\ }$ are independent Gaussian random vectors with mean ${\displaystyle {\bar {a}}_{i}}$ and covariance ${\displaystyle \Sigma _{i}\ }$ and ${\displaystyle p\geq 0.5}$. This problem can be expressed as the SOCP

minimize ${\displaystyle \ c^{T}x\ }$
subject to
${\displaystyle {\bar {a}}_{i}^{T}x+\Phi ^{-1}(p)\lVert \Sigma _{i}^{1/2}x\rVert _{2}\leq b_{i},\quad i=1,\dots ,m}$

where ${\displaystyle \Phi ^{-1}\ }$ is the inverse normal cumulative distribution function.[1]

## Example: Stochastic second-order cone programming

We refer to second-order cone programs as deterministic second-order cone programs since data defining them are deterministic. Stochastic second-order cone programs[2] is a class of optimization problems that defined to handle uncertainty in data defining deterministic second-order cone programs.