socp

SOCP, or second-order cone programming, is a class of convex optimization problems characterized by linear objectives and constraints that include second-order (quadratic) cone inequalities. It sits between linear programming and semidefinite programming in expressiveness and is widely used for problems involving Euclidean norms and distances.

A standard form of an SOCP minimizes a linear objective subject to linear equalities and a set of second-order cone constraints. A typical representation is: minimize c^T x subject to ||F_i x + g_i||_2 ≤ h_i^T x + k_i for i = 1,...,m, along with optional linear equalities A x = b. The second-order cone, also called the quadratic cone, is Q_n = { (t, z) ∈ R × R^n : ||z||_2 ≤ t }. Each conic constraint enforces that a vector lies inside a cone, which corresponds to a norm-bound inequality.

SOCPs are solvable in polynomial time by interior-point methods and can handle large-scale, sparse problems efficiently. They are more expressive than linear programs but less general than semidefinite programs; many problems that involve l2-norms, Euclidean distances, or robust constraints can be formulated as SOCPs and solved using specialized solvers.

Applications span engineering, finance, and data science. Notable areas include signal processing, control and robotics, wireless communications, robust statistics, portfolio optimization with risk constraints, and certain machine learning tasks such as regularized regression with conic constraints.

Most SOCPs can be solved by modern optimization software, often by reformulating them into conic or SDP problems and applying interior-point or first-order methods optimized for large-scale instances.