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 $g\in {\mathbb {R}}^{p}$ . ( $\lVert x\rVert _{2}$ is the Euclidean norm and ( $T$ indicates Transpose) 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.

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.

Example: Quadratic constraint

Consider a quadratic constraint of the form

x^{T}A^{T}Ax+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}^{T}x\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 $p\geq 0.5$ . 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
FICO Xpress	commercial
Gurobi	commercial	parallel SOCP barrier algorithm
MOSEK	commercial
ECOS	open-source	embedded SOCP solver

References

1 2 Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011.
↑ 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.

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[boyd-1] 1 2 Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011.

[Alzalg-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.