# 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 $\ f^T x \$
subject to
$\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m$
$Fx = g \$

where the problem parameters are $f \in \mathbb{R}^n, \ A_i \in \mathbb{R}^{{n_i}\times n}, \ b_i \in \mathbb{R}^{n_i}, \ c_i \in \mathbb{R}^n, \ d_i \in \mathbb{R}, \ F \in \mathbb{R}^{p\times n}$, and $g \in \mathbb{R}^p$. Here $x\in\mathbb{R}^n$ is the optimization variable.[1]

When $A_i = 0$ for $i = 1,\dots,m$, the SOCP reduces to a linear program. When $c_i = 0$ for $i = 1,\dots,m$, the SOCP is equivalent to a convex quadratically constrained quadratic program. 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

$x^T A^T A x + b^T x + c \leq 0.$

This is equivalent to the SOC constraint

$\left\| \begin{matrix} (1 + b^T x +c)/2\\ Ax \end{matrix} \right\|_2 \leq (1 - b^T x -c)/2.$

## Example: Stochastic programming

Consider a stochastic linear program in inequality form

minimize $\ c^T x \$
subject to
$P(a_i^T(x) \geq b_i) \geq p, \quad i = 1,\dots,m$

where the parameters $a_i \$ are independent Gaussian random vectors with mean $\bar{a}_i$ and covariance $\Sigma_i \$ and $p\geq0.5$. This problem can be expressed as the SOCP

minimize $\ c^T x \$
subject to
$\bar{a}_i^T (x) + \Phi^{-1}(1-p) \lVert \Sigma_i^{1/2} x \rVert_2 \geq b_i , \quad i = 1,\dots,m$

where $\Phi^{-1} \$ is the inverse error function.[1]