Leveraging Tucker low-rank decomposition for dimensionality reduction and over-fitting mitigation, this study investigates stochastic tensor Tucker optimization problems subject to deterministic constraints, with an emphasis on designing a Stochastic Proximal Alternating Direction Method of Multipliers (SPADMM). Instead of treating the underlying polynomial structures as general smooth nonlinear functions, we introduce block conservative functions based on the tensor Tucker decomposition, and further analyse the convergence rate and high probability performance for the proposed stochastic algorithm. Both numerical tests and background/foreground separation experiments demonstrate that our approach is promising in time efficiency and solution accuracy.
In this paper, we develop a modified inertial triple self-adaptive algorithm that integrates the inertial method, the subgradient extragradient method, and the hybrid steepest descent method for solving bilevel variational inequality problems, where the upper-level problem involves a strongly monotone variational inequality problem and the lower-level problem comprises a quasimonotone variational inequality problem with split variational inequalities and fixed-point constraints. A distinctive feature of our approach is a triple self-adaptive step-size mechanism that dynamically adjusts parameters without requiring prior estimation of operator norms or Lipschitz constants. We rigorously establish the strong convergence theory for the algorithm under the bilevel framework. Experimental results on two numerical examples and signal recovery applications, including the modified LASSO problem, demonstrate that the proposed algorithm achieves superior performance compared to existing methods in terms of both computational efficiency and practical applicability.
