Second-order cone programming

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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.

Example: Quadratic constraint[edit]

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[edit]

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]

Solvers and scripting (programming) languages[edit]

Name License Brief info
Xpress commercial from 7.6 release
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
SeDuMi GPL v3 Matlab package with primal–dual interior point methods[2]
SDPT3 GPL v2 Matlab package with primal–dual interior point methods[3] [4]
OpenOpt BSD universal cross-platform numerical optimization framework, see its SOCP page and other problems involved. Uses NumPy arrays and SciPy sparse matrices.

References[edit]

  1. ^ a b Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011. 
  2. ^ Sturm, Jos F. (1999). "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones". Optimization Methods and Software. 11-12: 625–653. 
  3. ^ Toh, K.C.; M.J. Todd, and R.H. Tutuncu (1999). "SDPT3 - a Matlab software package for semidefinite programming". Optimization Methods and Software 11: 545–581. doi:10.1080/10556789908805762. 
  4. ^ Tutuncu, R.H.; K.C. Toh, and M.J. Todd (2003). "Solving semidefinite-quadratic-linear programs using SDPT3". Mathematical Programming. Ser. B, 95: 189–217.