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]
The standard or unit second-order cone of dimension is defined as
.
The second-order cone is also known by quadratic cone or ice-cream cone or Lorentz cone. The standard 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
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]
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.[10]
Other examples
Other modeling examples are available at the MOSEK modeling cookbook.[11]