SeDuMi

SeDuMi is a MATLAB toolbox for optimization over symmetric cones. It solves convex optimization problems that can be written in conic form, including semidefinite programming (SDP), second-order cone programming (SOCP), and linear programming (LP). The solver employs interior-point methods based on a homogeneous self-dual embedding, which reformulates a primal-dual pair into a single feasible problem and can provide certificates of optimality or infeasibility.

A typical problem format is minimize c^T x subject to A x = b and x in K, where K is a product of cones such as the nonnegative orthant, second-order cones, and the cone of positive semidefinite matrices. SeDuMi handles both sparse and dense data and is implemented in MATLAB, with options to leverage external linear algebra packages for performance.

SeDuMi is widely used in MATLAB-based convex optimization workflows and is commonly invoked directly or through higher-level toolboxes such as CVX and YALMIP. It provides primal and dual solutions, the duality gap, feasibility status, and solver statistics (iterations and timing). While robust and accessible, SeDuMi is generally outperformed on very large-scale problems by specialized solvers; for many moderate-size problems it remains a convenient, reliable choice.

History and availability: SeDuMi originated in the early 2000s as a MATLAB implementation for optimization over symmetric cones. It is distributed with documentation and example scripts and has been a staple within the MATLAB optimization community, particularly as a backend solver for CVX.