Second Order Dynamical System for Solving Tensor Split Feasibility Problem

What the paper says

(Communicated by Zheng-Hai Huang) The tensor split feasibility problem (TSFP), as a high-order extension of the classical split feasibility problem (SFP), has important applications in areas such as information retrieval. Its objective is to find a vector $x \in \mathbb{R}^n$ such that $x\in C~~\text{and}~~\mathcal A x^{m-1}\in Q,$ where $C \subseteq \mathbb{R}^n$ and $Q \subseteq \mathbb{R}^p$ are nonempty closed convex sets, and $\mathcal{A}$ is a real tensor of order $m$ with dimensions $p \times n \times \cdots \times n$. Existing numerical methods, including the projection algorithm and the Levenberg-Marquardt (LM) algorithm, suffer from several limitations: their performance often depends on the initial point lying within a traditional convergence region, and the inner--outer iterative structure may lead to increased computational cost. To enhance the efficiency of solving TSFP, we reformulate the problem as the merit function $p(x)=\tfrac12\|P_C(x)-x\|^2+\tfrac12\|P_Q(\mathcal{A}x^{m-1})-\mathcal{A}x^{m-1}\|^2,$ whose gradient is $g(x)=(I-P_C)x+J(x)^\top (I-P_Q)\mathcal{A}x^{m-1},\qquad$ $J(x)=(m-1)\mathcal{A}x^{m-2}.$ This leads to the fixed-point formulation $x=P_\Omega(x-\alpha g(x)),$ where $\Omega$ is a closed convex set containing the TSFP solution set. Based on this fixed-point formulation, we introduce a novel second order dynamical system based on the gradient projection method, which yields the following continuous-time algorithm $$\begin{cases}\ddot{x}(t)+\gamma(t)\dot{x}(t)+\lambda(t)\bigl(x(t)-P_\Omega(x(t)-\alpha g(x(t)))\bigr)=0,\\[2mm]x(0)=u_0,\quad \dot{x}(0)=v_0,\end{cases}$$ where $\alpha>0$, $\gamma(t)$ and $\lambda(t)$ are Lebesgue measurable functions. In the theoretical analysis, we show that the merit function $p(x)$, its gradient $g(x)$, the tensor mappings $\mathcal{A}x^{m}$, $\mathcal{A}x^{m-1}$, and the Jacobian $J(x)$ are Lipschitz continuous on bounded closed sets, thereby ensuring the well-posedness of the dynamical system. It is further proved that the proposed algorithm has a unique solution under mild conditions. Furthermore, the corresponding trajectory of the algorithm always converges to the unique solution. Compared with up-to-date methods, numerical experiments are given to verify the effectiveness and competitiveness of the proposed algorithm. The experimental results show that the proposed algorithm consistently outperforms the projection algorithm and the LM algorithm in terms of computational efficiency.