Haitao Che

(Communicated by Guanglu Zhou) Consider the linear equation $Kx = y,$ (1) where $x\in\mathbb{R}^n $ is the original signal or image, $K:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is a degradation operator, and $y\in\mathbb{R}^m$ is the observed signal or image. It has some important applications to signal processing, non-destructive testing, etc. The typical $\ell_1$-norm sparse regularization method does not have beneficial stability for ill-posed problems with large condition number. What's more, in many applications, the solution $x$ of the question (1) is not really sparse. To overcome the shortcomings of typical sparsity regularization, some researchers have focused on the elastic-net regularization problem, that is, $\min_{x\in \mathbb{R}^n} J(x)=\frac{1}{2}\|Kx-y\|^2+\alpha\|x\|_{1}+\frac{\beta}{2}\|x\|^2,$ (2) where $\alpha>0, \beta>0, $ $\|\cdot\|$ represents $\ell_2$-norm, $\|\cdot\|_1$ represents $\ell_1$-norm. As a species of multi-parameter Tikhonov regularization, elastic-net regularization is more applicable to ill-conditioned issues. Statistically speaking, elastic-net regularization has superior steadiness than the typical regularization. There is an additional $\ell_2$-norm in elastic-net regularization compared to the classical $\ell_1$-norm sparsity regularization. Elastic-net regularization has the advantages of both $\ell_2$-regularization and $\ell_1$-regularization. Even though there exist various approaches to solve the elastic-net regularization problem which mainly focus on the convergence of the algorithm, not much work has been carried out in the convergence time of the algorithm. Therefore, it is necessary to devise a method to estimate the convergence time. For (2), we can relax the problem to be the following constrained optimization problem min $g(x)=\frac{1}{2} \|Kx-y\|^2+\frac{\beta}{2}\|x\|^2$ s.t. $x\in\Theta,$ (3) where $\Theta=\{x\in\mathbb{R}^n|\|x\|_{1}\leq r\}, r>0,$ and transform the problem (3) into the fixed point equation form $x=P_\Theta(x-\lambda K^T(Kx-y)-\lambda\beta x),$ (4) where the stepsize $\lambda>0$. Obviously, the solution of problem (1) can be obtained by taking the solution of problem (3) when $\beta\rightarrow0$. Based on (4), we propose the following new continuous-time projection algorithm for solving the elastic-net regularization $\dot{x}(t)=-\varphi(x)(x(t)-z(x(t))),$ (5) where $\varphi(x)=\left\{\begin{array}{l}\frac{k_1}{\|z(x(t))-x(t)\|^{1-\alpha_1}}+\frac{k_2}{\|z(x(t))-x(t)\|^{1-\alpha_2}}+\frac{k_3}{\|z(x(t))-x(t)\|},\mathrm{if}~ x\in\mathbb{R}^n\setminus \{x\in \mathbb{R}^n:x=z(x)\},\\0,~~~~~~~~~\mathrm{otherwise}, \end{array}\right. $ $z(x(t))=P_\Theta(x(t)-\lambda(K^TKx(t)-K^Ty)-\lambda\beta x(t)),$ $k_1>0, k_2>0, k_3>0,\alpha_1\in(0,1), \alpha_2>1, \lambda>0$ and $\beta>0$. The equivalence between the equilibrium point of (5) and the solution of (4) is displayed. Moreover, we prove that for every $\beta>0$ and $\lambda\in(0,\frac{2\beta}{L^2})$, if $x$ is an equilibrium point of (5), then for all $t\in[0,+\infty)$, the solution of (5) exists uniquely for any given initial value. Furthermore, the convergence time of the proposed method can be reckoned by the setting time $T(x(0))\leq T_{\max}=\frac{1}{d_1(1-p_1)}+\frac{1}{d_2(p_2-1)},$ where $k_1>0, k_2>0, k_3>0, \alpha_1\in(0,1), \alpha_2>1, L=\|K\|^2+\beta, \lambda\in(0,\frac{2\beta}{L^2}), \Phi=\sqrt{\frac{1}{1-\lambda^2 L^2+2\lambda\beta}}\in (0,1),$ $p_1:=\frac{1+\alpha_1}{2},$ $p_2:=\frac{1+\alpha_2}{2},$ $d_1=k_1(1-\Phi)\frac{2^{p_1}}{(\frac{4\beta}{4\beta-\lambda L^2})^{1-\alpha_1}}>0,$ $d_2=2^{p_1}(1-\Phi)^{\alpha_2}k_2>0. $ To validate the convergence and effectiveness of the proposed algorithm, we apply our algorithm to some numerical examples of signal processing and image restoration. Compared with state-of-the-art methods in the literature such as CAPPA method, THZF method and PNN algorithm, our algorithm is more effective to calculate the mean squared error (MSE) in signal processing. Furthermore, our algorithm shows the better performance to achieve the signal-to-noise ratio (SNR), the structural similarity index (SSIM) and the relative error (RE) in image restoration.

(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.