# Second-order cone programming

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

minimize ${\displaystyle \ f^{T}x\ }$
subject to
${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots ,m}$
${\displaystyle Fx=g\ }$

where the problem parameters are ${\displaystyle f\in \mathbb {R} ^{n},\ A_{i}\in \mathbb {R} ^{{n_{i}}\times n},\ b_{i}\in \mathbb {R} ^{n_{i}},\ c_{i}\in \mathbb {R} ^{n},\ d_{i}\in \mathbb {R} ,\ F\in \mathbb {R} ^{p\times n}}$, and ${\displaystyle g\in \mathbb {R} ^{p}}$. ${\displaystyle x\in \mathbb {R} ^{n}}$ is the optimization variable. ${\displaystyle \lVert x\rVert _{2}}$ is the Euclidean norm and ${\displaystyle ^{T}}$ indicates transpose.[1] The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function ${\displaystyle (Ax+b,c^{T}x+d)}$ to lie in the second-order cone in ${\displaystyle \mathbb {R} ^{n_{i}+1}}$.[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]

## Second-order cone

The standard or unit second-order cone of dimension ${\displaystyle n+1}$ is defined as

${\displaystyle {\mathcal {C}}_{n+1}=\left\{{\begin{bmatrix}x\\t\end{bmatrix}}{\Bigg |}x\in \mathbb {R} ^{n},t\in \mathbb {R} ,||x||_{2}\leq t\right\}}$.

The second-order cone is also known by quadratic cone, ice-cream cone, or Lorentz cone. The second-order cone in ${\displaystyle \mathbb {R} ^{3}}$ is ${\displaystyle \left\{(x,y,z){\Big |}{\sqrt {x^{2}+y^{2}}}\leq z\right\}}$.

The set of points satisfying a second-order cone constraint is the inverse image of the unit second-order cone under an affine mapping:

${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}\Leftrightarrow {\begin{bmatrix}A_{i}\\c_{i}^{T}\end{bmatrix}}x+{\begin{bmatrix}b_{i}\\d_{i}\end{bmatrix}}\in {\mathcal {C}}_{n_{i}+1}}$

and hence is convex.

The second-order cone can be embedded in the cone of the positive semidefinite matrices since

${\displaystyle ||x||\leq t\Leftrightarrow {\begin{bmatrix}tI&x\\x^{T}&t\end{bmatrix}}\succcurlyeq 0,}$

i.e., a second-order cone constraint is equivalent to a linear matrix inequality (Here ${\displaystyle M\succcurlyeq 0}$ means ${\displaystyle M}$ is semidefinite matrix). Similarly, we also have,

${\displaystyle \lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}\Leftrightarrow {\begin{bmatrix}(c_{i}^{T}x+d_{i})I&A_{i}x+b_{i}\\(A_{i}x+b_{i})^{T}&c_{i}^{T}x+d_{i}\end{bmatrix}}\succcurlyeq 0}$.

## Relation with other optimization problems

A hierarchy of convex optimization problems. (LP: linear program, QP: quadratic program, SOCP second-order cone program, SDP: semidefinite program, CP: cone program.)

When ${\displaystyle A_{i}=0}$ for ${\displaystyle i=1,\dots ,m}$, the SOCP reduces to a linear program. When ${\displaystyle c_{i}=0}$ for ${\displaystyle i=1,\dots ,m}$, 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.[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

Consider a quadratic constraint of the form

${\displaystyle x^{T}Ax+b^{T}x+c\leq 0.}$

This is equivalent to the SOC constraint

${\displaystyle \lVert A^{1/2}x+{\frac {1}{2}}A^{-1/2}b\rVert \leq \left({\frac {1}{4}}b^{T}A^{-1}b-c\right)^{\frac {1}{2}}}$

### Stochastic linear programming

Consider a stochastic linear program in inequality form

minimize ${\displaystyle \ c^{T}x\ }$
subject to
${\displaystyle \mathbb {P} (a_{i}^{T}x\leq b_{i})\geq p,\quad i=1,\dots ,m}$

where the parameters ${\displaystyle a_{i}\ }$ are independent Gaussian random vectors with mean ${\displaystyle {\bar {a}}_{i}}$ and covariance ${\displaystyle \Sigma _{i}\ }$ and ${\displaystyle p\geq 0.5}$. This problem can be expressed as the SOCP

minimize ${\displaystyle \ c^{T}x\ }$
subject to
${\displaystyle {\bar {a}}_{i}^{T}x+\Phi ^{-1}(p)\lVert \Sigma _{i}^{1/2}x\rVert _{2}\leq b_{i},\quad i=1,\dots ,m}$

where ${\displaystyle \Phi ^{-1}(\cdot )\ }$ 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

AMPL commercial An algebraic modeling language with SOCP support
Artelys Knitro commercial
CPLEX commercial
FICO Xpress commercial
Gurobi commercial parallel SOCP barrier algorithm
MATLAB commercial The coneprog function solves SOCP problems[7] using an interior-point algorithm[8]
MOSEK commercial parallel interior-point algorithm
NAG Numerical Library commercial General purpose numerical library with SOCP solver

## References

1. ^ a b c Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved July 15, 2019.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ Scheiderer, Claus (2020-04-08). "Second-order cone representation for convex subsets of the plane". arXiv:2004.04196 [math.OC].
6. ^ Scheiderer, Claus (2018). "Spectrahedral Shadows". SIAM Journal on Applied Algebra and Geometry. 2 (1): 26–44. doi:10.1137/17M1118981. ISSN 2470-6566.
7. ^ "Second-order cone programming solver - MATLAB coneprog". MathWorks. 2021-03-01. Retrieved 2021-07-15.
8. ^ "Second-Order Cone Programming Algorithm - MATLAB & Simulink". MathWorks. 2021-03-01. Retrieved 2021-07-15.