A Triangular Log-Det PDHG for Semidefinite Programming: Complementarity Selection and Global Linear Convergence

Semidefinite programming (SDP) is an important class of convex optimization problems with wide applications in practice. In this paper, we study a solution-selection question for first-order methods for solving SDPs: Can a first-order method provably select a strictly complementary solution when one exists, and a maximally complementary solution otherwise? We propose Triangular Log-Det PDHG (TriLog-PDHG), a primal-dual hybrid gradient method equipped with a triangular log-determinant Bregman kernel. We show that TriLog-PDHG converges to a KKT point for any fixed positive shift and to a maximally complementary solution for every sufficiently small positive shift. When a strictly complementary solution exists, the unshifted method provably selects a strictly complementary KKT point and enjoys a global linear convergence rate. Under the same condition, we further develop an averaged and adaptively restarted variant that accelerates convergence. Numerical experiments on classical examples and SDPLIB instances illustrate the complementarity-selection and convergence properties predicted by the theory.