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A Tuning-Free Primal-Dual Splitting Algorithm for Large-Scale Semidefinite Programming
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This paper proposes and analyzes a tuning-free variant of Primal-Dual Hybrid Gradient (PDHG), and investigates its effectiveness for solving large-scale semidefinite programming (SDP). The core idea is based on the combination of two seemingly unrelated results: (1) the equivalence of PDHG and Douglas-Rachford splitting (DRS); (2) the asymptotic convergence of non-stationary DRS. This combination provides a unified approach to analyze the convergence of generic adaptive PDHG, including the proposed tuning-free algorithm and various existing ones. Numerical experiments are conducted to show the performance of our algorithm, highlighting its superior convergence speed and robustness in the context of SDP.
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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.
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