SOCP stands for Second Order Cone Programing.

Second Order Cone Programing is a class of convex optimization problems that lies between Linear Programming and Semidefinite Programming.

Such problems consist in minimizing a linear objective under second order constraints:

Min_x f^Tx    subject to    ||A_ix + b_i||≤ c_i^T x + d_i ,  i=1,...,N

SOCP includes the family of Linear Programs, which can be seen by taking Ai the zero-matrix and bi the zero-vector. SOCP also includes many optimization problems, like quadratic programs and problems with hyperbolic constraints.

Second order cone programs have convex constraints hence can be solved via the Interior Point methods, developed by Khachian and Karmarkar for linear programming in 1984 and generalized by Nesterov and Nemirovskii to convex optimization problems in 1994.

SOCP has many applications in the finance industry, mainly in portfolio selection issues. SOCP covers a wide range of optimization problems and opened the way to a new family of investment strategies, by introducing advanced risk measures in the allocation process itself.