SDPsolvers

SDPsolvers are software packages designed to solve semidefinite programming (SDP) problems, a class of convex optimization where the decision variable is a symmetric matrix constrained to be positive semidefinite. An SDP typically minimizes a linear objective in terms of a matrix variable X, subject to linear equalities or inequalities and the constraint X ≽ 0. Problems can be formulated in standard forms that emphasize the matrix positivity constraint and linear matrix inequalities.

Most SDPsolvers implement primal-dual interior-point methods, often with enhancements for sparse, structured, or large-scale problems. Some solvers also provide first-order or augmented Lagrangian methods for greater scalability, trading off precision for speed. They generally return an optimal matrix X, dual variables, and convergence certificates that attest to optimality within a specified tolerance. The numerical behavior of SDPsolvers can be influenced by problem conditioning, scaling, and sparsity pattern.

SDPsolvers are commonly accessed through modeling tools and programming interfaces that translate high-level problem descriptions into solver calls. Popular modeling environments include CVX, YALMIP, and JuMP, which support multiple solvers and facilitate switching among backends without rewriting problem code. Notable SDPsolvers include SeDuMi, SDPT3, SDPA, MOSEK, and SCS, among others; these range from open-source to commercial offerings and differ in performance characteristics and supported features.

Applications of SDPsolvers span control theory, quantum information, combinatorial optimization via SDP relaxations, structural design, and certain machine learning tasks such as kernel learning and matrix completion. The field continues to evolve with advances in algorithms for large-scale and structured SDPs, as well as enhanced numerical robustness and usability.