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<p>In this study, we consider the complex tensor equation. We propose a linearized alternating direction method of multipliers (LADMM) through the Wirtinger calculus technique. Compared to the alternating direction method of multipliers (ADMM), our LADMM algorithm is more robust and does not need to solve the linear equation in each iteration. By applying the tensor-train (TT) decomposition, we only need to compute matrix products and third-order tensor products instead of high-order tensor products. Under mild assumptions, we prove that all limit points of the sequences generated by our LADMM algorithm satisfy the corresponding Karush-Kuhn-Tucker (KKT) conditions, and the residuals of the sequences converge to 0. Moreover, we solve US-eigenvalue problems, higher-order Markov chain problems, and general constrained tensor equations by using our LADMM algorithm. Finally, we conduct some numerical experiments to illustrate the effectiveness of our LADMM algorithm.</p>

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A Linearized Alternating Direction Method of Multipliers for Solving Complex Semi-Symmetric Tensor Equations and its Applications

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Shenghao Feng,
  • Yimin Wei,
  • Renjie Xu,
  • Eric King-wah Chu

摘要

In this study, we consider the complex tensor equation. We propose a linearized alternating direction method of multipliers (LADMM) through the Wirtinger calculus technique. Compared to the alternating direction method of multipliers (ADMM), our LADMM algorithm is more robust and does not need to solve the linear equation in each iteration. By applying the tensor-train (TT) decomposition, we only need to compute matrix products and third-order tensor products instead of high-order tensor products. Under mild assumptions, we prove that all limit points of the sequences generated by our LADMM algorithm satisfy the corresponding Karush-Kuhn-Tucker (KKT) conditions, and the residuals of the sequences converge to 0. Moreover, we solve US-eigenvalue problems, higher-order Markov chain problems, and general constrained tensor equations by using our LADMM algorithm. Finally, we conduct some numerical experiments to illustrate the effectiveness of our LADMM algorithm.