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 linear program. 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.

Example: Quadratic constraint

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 linear programming

Consider a stochastic linear program in inequality form

minimize $\ c^T x \$

subject to

$P(a_i^Tx \leq b_i) \geq p, \quad i = 1,\dots,m$

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

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

where $\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.

Solvers and scripting (programming) languages

Name	License	Brief info
AMPL	commercial	An algebraic modeling language with SOCP support
CPLEX	commercial
ECOS	GPL v3	SOCP solver for embedded applications
Gurobi	commercial	parallel SOCP barrier algorithm
JOptimizer	Apache License	Java library for convex optimization (open source)
MOSEK	commercial
OpenOpt	BSD	universal cross-platform numerical optimization framework, see its SOCP page and other problems involved. Uses NumPy arrays and SciPy sparse matrices.
SCS	MIT License	C library that solves large-scale convex cone problems
SDPT3	GPL v2	Matlab package with primal–dual interior point methods[2][3][4][5][6]
Xpress	commercial	from 7.6 release

References

1. Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011.
2. Alzalg, Baha (2012). "Stochastic second-order cone programming: Application models". Applied Mathematical Modelling 36 (10): 5122–5134. doi:10.1016/j.apm.2011.12.053.
3. Toh, K.C.; M.J. Todd; R.H. Tutuncu (1999). "SDPT3 - a Matlab software package for semidefinite programming". Optimization Methods andSoftware 11: 545–581. doi:10.1080/10556789908805762.
4. Tutuncu, R.H.; K.C. Toh; M.J. Todd (2003). "Solving semidefinite-quadratic-linear programs using SDPT3". Mathematical Programming. B 95: 189–217. doi:10.1007/s10107-002-0347-5.
5. |SeDuMi||GPL v3||Matlab package with primal–dual interior point methods
6. Sturm, Jos F. (1999). "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones". Optimization Methods and Software. 11-12: 625–653.