| # Standard Forms of SOCP SOCP > SOC inequalities | Standard forms | Group sparsity | Applications Second-order cone programs: standard forms Quadratically constrained quadratic programming ## Second-order cone programs: standard forms ### Inequality form A second-order cone program (or SOCP, for short) is an optimization problem of the form where 's are given matrices, , vectors, and 's scalars. The problem is convex, since the constraint functions of the corresponding standard form are. Examples: Linear program as SOCP. Facility location. Robust least-squares. Separation of ellipsoids. ### Conic form We can put the above problem in the so-called ‘‘conic’’ format Since the cones are convex, and the mappings are affine, the feasible set is convex. ### Rotated second-order cone constraints Since the rotated second-order cone can be expressed as some linear transformation of an ordinary second-order cone, we can include rotated second-order cone constraints, as well as ordinary linear inequalities or equalities, in the formulation. This allows to formulate LPs and QPs as special cases of SOCP. Examples: Quadratic program as SOCP. Logarithmic Chebyschev approximation ## Quadratically constrained quadratic programming ### Definition A quadratically constrained quadratic programming (QCQP for short) is a problem of the form where , . This condition ensures that the problem is convex. QCQPs contain LPs and QPs as special case. Examples: ### SOCP formulation We can formulate QCQPs as SOCPs, by introducing new variables and affine equality constraints. (Proof).