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]

Relation with other optimization problems

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

Quadratic constraint

Consider a quadratic constraint of the form

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

This is equivalent to the SOC constraint

${\displaystyle \left\|{\begin{matrix}(1+b^{T}x+c)/2\\Ax\end{matrix}}\right\|_{2}\leq (1-b^{T}x-c)/2.}$

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

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

1. 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. 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. 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.