A secondorder 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.
Contents

Example: Quadratic constraint 1

Example: Stochastic linear programming 2

Example: Stochastic secondorder cone programming 3

Solvers and scripting (programming) languages 4

References 5
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 secondorder cone programming
We refer to secondorder cone programs as deterministic secondorder cone programs since data deﬁning them are deterministic. Stochastic secondorder cone programs^{[2]} is a class of optimization problems that deﬁned to handle uncertainty in data deﬁning deterministic secondorder 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 crossplatform numerical optimization framework, see its SOCP page and other problems involved. Uses NumPy arrays and SciPy sparse matrices.

SDPT3

GPL v2

Matlab package with primal–dual interior point methods^{[3]}^{[2]}^{[4]}^{[5]}^{[6]}

Xpress

commercial

from 7.6 release

References

^ ^{a} ^{b} Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press.

^ ^{a} ^{b} Alzalg, Baha (2012). "Stochastic secondorder cone programming: Application models". Applied Mathematical Modeling 36 (10): 5122–5134.

^ Toh, K.C.; M.J. Todd; R.H. Tutuncu (1999). "SDPT3  a Matlab software package for semidefinite programming". Optimization Methods andSoftware 11: 545–581.

^ Tutuncu, R.H.; K.C. Toh; M.J. Todd (2003). "Solving semidefinitequadraticlinear programs using SDPT3". Mathematical Programming. B 95: 189–217.

^ SeDuMiGPL v3Matlab package with primal–dual interior point methods

^ Sturm, Jos F. (1999). "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones". Optimization Methods and Software. 1112: 625–653.
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