PDHG converges locally linearly for SDP under strict complementarity or primal-dual nondegeneracy, and can converge sublinearly when both fail.
Interior point methods in semidefinite programming with applications to combinatorial optimization.SIAM Journal on Optimization, 5(1):13–51, 1995
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Local Linear Convergence of the Primal-Dual Hybrid Gradient Method for Semidefinite Programming
PDHG converges locally linearly for SDP under strict complementarity or primal-dual nondegeneracy, and can converge sublinearly when both fail.