Second-order cone programming

A second-order cone program (SOCP) is a convex optimization problem of the form where the problem parameters areis the optimization variable.is the Euclidean norm and[1] The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine functionto 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] The standard or unit second-order cone of dimensionThe second-order cone is also known by quadratic cone or ice-cream cone or Lorentz cone.The standard second-order cone inThe set of points satisfying a second-order cone constraint is the inverse image of the unit second-order cone under an affine mapping: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 (Hereis semidefinite matrix)., the SOCP reduces to a linear program., 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] Any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP,[8].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).[9] Consider a convex quadratic constraint of the form This is equivalent to the SOCP constraint Consider a stochastic linear program in inequality form where the parametersare independent Gaussian random vectors with meanThis problem can be expressed as the SOCP whereis the inverse normal cumulative distribution function.[1] 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 modeling examples are available at the MOSEK modeling cookbook.