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.
This paper investigates optimality conditions for generalized Nash equilibrium problems (GNEPs) where the objective and constraint functions are quasidifferentiable. Quasidifferentiability allows for a more flexible analysis of equilibrium problems with nonconvex and nonsmooth structures. We derive first-order necessary optimality conditions in terms of quasidifferentials, extending classical Karush-Kuhn-Tucker (KKT) conditions to this broader setting. As these optimality conditions are formulated in terms of the marginal functions associated with each player's optimization subproblem, we conduct a rigorous sensitivity analysis to characterize their properties. Additionally, under appropriate generalized convexity assumptions, we establish sufficient conditions for optimality. Our results contribute to the theory of nonsmooth equilibrium problems by providing a unified framework for analysing GNEPs with nonstandard differentiability properties. The applicability of the derived conditions is illustrated through examples.
We propose a relative-error inexact Spingarn's operator splitting algorithm for solving monotone inclusion problems for finite sums of maximal monotone operators defined in a real Hilbert space. In contrast to projective-splitting type methods, which iterate by means of projections onto separating hyperplanes, the proposed algorithm uses extragradient steps, in the spirit of the Korpelevich and HPE-type methods. We analyze the iteration-complexity of the algorithm, establishing O(1/k) $ O(1/\sqrt {k}) $ pointwise rate and improved O(1/k) $ O(1/k) $ ergodic rate. Additionally, we prove the weak convergence of the iterates to a solution, if any.
In this paper, we propose a hybrid-adaptive accelerated Douglas-Rachford splitting method (HAA-DRSM) for solving absolute value equations of the form Ex-|x|-p=0 $ Ex-|x|-p=0 $ . Unlike existing inertial acceleration techniques, HAA-DRSM employs a hybrid strategy that combines convex combination, inertial extrapolation, and adaptive regulation. By leveraging multiple past iterates, it generates the inertial point via two distinct alternating schemes. Under mild assumptions, including & Vert;E-1 & Vert;<= 1 $ \|E<^>{-1}\| \leq 1 $ , we establish global convergence and convergence-rate results for HAA-DRSM. Numerical experiments are conducted to demonstrate the effectiveness and competitiveness of HAA-DRSM.
