Second-order cone programming

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

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] Applications in quantitative finance include portfolio optimization; some market impact constraints, because they are not linear, cannot be solved by quadratic programming but can be formulated as SOCP problems.[5][6][7]

Second-order cone[edit]

The standard or unit second-order cone of dimension is defined as


The second-order cone is also known by quadratic cone, ice-cream cone, or Lorentz cone. The second-order cone in is .

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

and hence is convex.

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

i.e., a second-order cone constraint is equivalent to a linear matrix inequality (Here means is semidefinite matrix). Similarly, we also have,


Relation with other optimization problems[edit]

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

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,[8] 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.[9]


Quadratic constraint[edit]

Consider a convex quadratic constraint of the form

This is equivalent to the SOCP constraint

Stochastic linear programming[edit]

Consider a stochastic linear program in inequality form

subject to

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

subject to

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

Stochastic second-order cone programming[edit]

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[edit]

Name License Brief info
AMPL commercial An algebraic modeling language with SOCP support
Artelys Knitro commercial
CPLEX commercial
FICO Xpress commercial
Gurobi commercial
MATLAB commercial The coneprog function solves SOCP problems[10] using an interior-point algorithm[11]
MOSEK commercial parallel interior-point algorithm
NAG Numerical Library commercial General purpose numerical library with SOCP solver


  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. S2CID 119324071.
  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. ^ "Solving SOCP" (PDF).
  6. ^ "portfolio optimization" (PDF).
  7. ^ Li, Haksun (16 January 2022). Numerical Methods Using Java: For Data Science, Analysis, and Engineering. APress. pp. Chapter 10. ISBN 978-1484267967.
  8. ^ Scheiderer, Claus (2020-04-08). "Second-order cone representation for convex subsets of the plane". arXiv:2004.04196 [math.OC].
  9. ^ Scheiderer, Claus (2018). "Spectrahedral Shadows". SIAM Journal on Applied Algebra and Geometry. 2 (1): 26–44. doi:10.1137/17M1118981. ISSN 2470-6566.
  10. ^ "Second-order cone programming solver - MATLAB coneprog". MathWorks. 2021-03-01. Retrieved 2021-07-15.
  11. ^ "Second-Order Cone Programming Algorithm - MATLAB & Simulink". MathWorks. 2021-03-01. Retrieved 2021-07-15.