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

- minimize
- subject to

where the problem parameters are , and . is the optimization variable.
is the Euclidean norm and indicates transpose.^{[1]} The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function to lie in the second-order cone in .^{[1]}

SOCPs can be solved by interior point methods^{[2]} and in general, can be solved more efficiently than semidefinite programming (SDP) problems.^{[3]} Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics.^{[4]}

## Relation with other optimization problems

When for , the SOCP reduces to a linear program. When for , 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.^{[4]} Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program.^{[4]} The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation.^{[3]} In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP,^{[5]} it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP.^{[6]}

## Examples

### Quadratic constraint

Consider a quadratic constraint of the form

This is equivalent to the SOC constraint

### Stochastic linear programming

Consider a stochastic linear program in inequality form

- minimize
- subject to

where the parameters are independent Gaussian random vectors with mean and covariance and . This problem can be expressed as the SOCP

- minimize
- subject to

where is the inverse normal cumulative distribution function.^{[1]}

### 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 are a class of optimization problems that are 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 |

Artelys Knitro | commercial | |

CPLEX | commercial | |

FICO Xpress | commercial | |

Gurobi | commercial | parallel SOCP barrier algorithm |

MOSEK | commercial | parallel interior-point algorithm |

NAG Numerical Library | commercial | General purpose numerical library with SOCP solver |

## References

- ^
^{a}^{b}^{c}Boyd, Stephen; Vandenberghe, Lieven (2004).*Convex Optimization*(PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved July 15, 2019. **^**Potra, lorian A.; Wright, Stephen J. (1 December 2000). "Interior-point methods".*Journal of Computational and Applied Mathematics*.**124**(1–2): 281–302. Bibcode:2000JCoAM.124..281P. doi:10.1016/S0377-0427(00)00433-7.- ^
^{a}^{b}Fawzi, Hamza (2019). "On representing the positive semidefinite cone using the second-order cone".*Mathematical Programming*.**175**(1–2): 109–118. arXiv:1610.04901. doi:10.1007/s10107-018-1233-0. ISSN 0025-5610. - ^
^{a}^{b}^{c}Lobo, Miguel Sousa; Vandenberghe, Lieven; Boyd, Stephen; Lebret, Hervé (1998). "Applications of second-order cone programming".*Linear Algebra and Its Applications*.**284**(1–3): 193–228. doi:10.1016/S0024-3795(98)10032-0. **^**Scheiderer, Claus (2020-04-08). "Second-order cone representation for convex subsets of the plane". arXiv:2004.04196 [math.OC].**^**Scheiderer, Claus (2018). "Spectrahedral Shadows".*SIAM Journal on Applied Algebra and Geometry*.**2**(1): 26–44. doi:10.1137/17M1118981. ISSN 2470-6566.