Second-order cone programming

A second-order cone program (SOCP) is a convex optimization problem of the form A second-order cone program (SOCP) is a convex optimization problem of the form where the problem parameters are f ∈ R n ,   A i ∈ R n i × n ,   b i ∈ R n i ,   c i ∈ R n ,   d i ∈ R ,   F ∈ R p × n {displaystyle fin mathbb {R} ^{n}, A_{i}in mathbb {R} ^{{n_{i}} imes n}, b_{i}in mathbb {R} ^{n_{i}}, c_{i}in mathbb {R} ^{n}, d_{i}in mathbb {R} , Fin mathbb {R} ^{p imes n}} , and g ∈ R p {displaystyle gin mathbb {R} ^{p}} . x ∈ R n {displaystyle xin mathbb {R} ^{n}} is the optimization variable. ‖ x ‖ 2 {displaystyle lVert x Vert _{2}} is the Euclidean norm and T {displaystyle ^{T}} indicates transpose.When A i = 0 {displaystyle A_{i}=0} for i = 1 , … , m {displaystyle i=1,dots ,m} , the SOCP reduces to a linear program. When c i = 0 {displaystyle c_{i}=0} for i = 1 , … , m {displaystyle i=1,dots ,m} , the SOCP is equivalent to a convex quadratically constrained linear program.The name arises from the constraints, which are equivalent to requiring the affine function ( A x + b , c T x + d ) {displaystyle (Ax+b,c^{T}x+d)} to lie in the second-order cone in R k + 1 {displaystyle mathbb {R} ^{k+1}} . 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 semi definite program. SOCPs can be solved by interior point methods. Consider a quadratic constraint of the form