Overall pressure ratio

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

minimize $f^{T}x$

subject to

$\Vert A_{i}x+b_i{\Vert }_2\le c_i^{T}x+d_i,i=1,\dots ,m$

$Fx=g$

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.

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.

Example: Quadratic constraint

Consider a quadratic constraint of the form

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

This is equivalent to the SOC constraint

${\Vert \begin{array}{c}(1+b^{T}x+c)/2\\ Ax\end{array}\Vert }_2\le (1-b^{T}x-c)/2.$

Example: Stochastic programming

Consider a stochastic linear program in inequality form

minimize $c^{T}x$

subject to

$P(a_i^T(x)\ge b_i)\ge p,i=1,\dots ,m$

where the parameters $a_i$ are independent Gaussian random vectors with mean ${\overline{a}}_i$ and covariance ${\mathrm{\Sigma }}_i$ and $p\ge 0.5$. This problem can be expressed as the SOCP

minimize $c^{T}x$

subject to

${\overline{a}}_i^T(x)+{\mathrm{\Phi }}^{-1}(1-p)\Vert {\mathrm{\Sigma }}_i^{1/2}x{\Vert }_2\ge b_i,i=1,\dots ,m$

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

Solvers and scripting (programming) languages

Name	License	Brief info
Xpress	commercial	from 7.6 release
CPLEX	commercial
Gurobi	commercial	parallel SOCP barrier algorithm
JOptimizer	Apache License	Java library for convex optimization (open source)
MOSEK	commercial
SeDuMi	GPL v3	Matlab package with primal–dual interior point methods
SDPT3	GPL v2	Matlab package with primal–dual interior point methods
OpenOpt	BSD	universal cross-platform numerical optimization framework, see its SOCP page and other problems involved. Uses NumPy arrays and SciPy sparse matrices.

References

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