Three‐variable symmetric alternating direction method of multipliers for non‐separable convex optimization

Convex optimization plays a fundamental role in many fields, and the Alternating Direction Method of Multipliers (ADMM) has demonstrated remarkable efficacy in solving two‐block separable problems. However, extending ADMM to more complex scenarios faces two distinct theoretical hurdles: The potential divergence of direct multi‐block extensions () even under strictly separable conditions, and the limitations of conventional variants in handling non‐separable objective functions featuring quadratically coupled variables. To simultaneously overcome these challenges, this paper proposes a Three‐Variable Symmetric Alternating Direction Method of Multipliers (TV‐SADMM). The proposed framework resolves the three‐block divergence issue through an additional multiplier update step integrated with a Prediction‐Correction framework, which ensures monotonic descent for the three‐variable architecture. Furthermore, it utilizes proximal regularization to enhance the convexity of the subproblems. Consequently, the proposed method rigorously guarantees theoretical convergence for three‐block non‐separable minimization. Numerical experiments on Direction of Arrival (DOA) estimation verify that TV‐SADMM achieves superior stability and faster convergence compared with conventional ADMM‐based methods.