A second-order cone program (SOCP) is a convex optimization problem of the form
minimize
fTx
subject to
\lVertAix+bi\rVert2\leq
| T | |
| c | |
| i |
x+di, i=1,...,m
Fx=g
where the problem parameters are
f\inRn, Ai\in
| {ni | |
| R |
x n}, bi\in
| ni | |
| R |
, ci\inRn, di\inR, F\inRp x
g\inRp
x\inRn
\lVertx\rVert2
T
The name "second-order cone programming" comes from the nature of the individual constraints, which are each of the form:
\lVertAx+b\rVert2\leqcTx+d
These each define a subspace that is bounded by an inequality based on a second-order polynomial function defined on the optimization variable
x
SOCPs can be solved by interior point methods[3] and in general, can be solved more efficiently than semidefinite programming (SDP) problems. 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]
The standard or unit second-order cone of dimension
n+1
l{C}n+1=\left\{\begin{bmatrix}x\ t\end{bmatrix}|x\inRn,t\inR,\|x\|2\leqt\right\}
The second-order cone is also known by the names quadratic cone, ice-cream cone[8], or Lorentz cone. For example, the standard second-order cone in
R3
\left\{(x,y,z)|\sqrt{x2+y2}\leqz\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:
\lVertAix+bi\rVert2\leq
| T | |
| c | |
| i |
x+di\Leftrightarrow\begin{bmatrix}Ai
| T | |
| \ c | |
| i |
\end{bmatrix}x+\begin{bmatrix}bi\ di\end{bmatrix}\in
| l{C} | |
| ni+1 |
and hence is convex.
The second-order cone can be embedded in the cone of the positive semidefinite matrices since
||x||\leqt\Leftrightarrow\begin{bmatrix}tI&x\ xT&t\end{bmatrix}\succcurlyeq0,
i.e., a second-order cone constraint is equivalent to a linear matrix inequality. The nomenclature here can be confusing; here
M\succcurlyeq0
M
xTMx\geq0forallx\inRn
which is not a linear inequality in the conventional sense.
Similarly, we also have,
\lVertAix+bi\rVert2\leq
| T | |
| c | |
| i |
x+di\Leftrightarrow\begin{bmatrix}
| T | |
| (c | |
| i |
x+di)I&Aix+bi\ (Aix+
| T | |
| b | |
| i) |
&
| T | |
| c | |
| i |
x+di\end{bmatrix}\succcurlyeq0
When
Ai=0
i=1,...,m
ci=0
i=1,...,m
Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint. 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. The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation.[9]
Any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP.[10] However, it is known that there exist convex semialgebraic sets of higher dimension that are not representable by SDPs; that is, there exist convex semialgebraic sets that can not be written as the feasible region of a SDP (nor, a fortiori, as the feasible region of a SOCP).[11]
Consider a convex quadratic constraint of the form
xTAx+bTx+c\leq0.
This is equivalent to the SOCP constraint
\lVertA1/2x+
| 1 | |
| 2 |
A-1/2b\rVert\leq\left(
| 1 | |
| 4 |
bTA-1b-c
| ||||
| \right) |
Consider a stochastic linear program in inequality form
minimize
cTx
subject to
| Tx | |
| P(a | |
| i |
\leqbi)\geqp, i=1,...,m
where the parameters
ai
\bar{a}i
\Sigmai
p\geq0.5
minimize
cTx
subject to
| T | |
| \bar{a} | |
| i |
x+\Phi-1(p)\lVert
| 1/2 | |
| \Sigma | |
| i |
x\rVert2\leqbi, i=1,...,m
where
\Phi-1( ⋅ )
We refer to second-order cone programsas 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.[12]
Other modeling examples are available at the MOSEK modeling cookbook.[13]
| Name | License | Brief info | |
|---|---|---|---|
| ALGLIB | free/commercial | A dual-licensed C++/C#/Java/Python numerical analysis library with parallel SOCP solver. | |
| AMPL | commercial | An algebraic modeling language with SOCP support | |
| Artelys Knitro | commercial | ||
| CPLEX | commercial | ||
| FICO Xpress | commercial | ||
| Gurobi Optimizer | commercial | ||
| MATLAB | commercial | The coneprog function solves SOCP problems[14] using an interior-point algorithm[15] | |
| MOSEK | commercial | parallel interior-point algorithm | |
| NAG Numerical Library | commercial | General purpose numerical library with SOCP solver |
