We propose a novel optimization for tensor completion by considering the minimal and the maximal Tucker rank. The objective function of the new programming is a weighted combination of the minimal and maximal nuclear norm of N-mode matrices instead of a weighted combination of nuclear norm of N-mode matrices, where the minimal nuclear norm is a nonconvex and nonsmooth function. The null space and the restricted isometry properties are discussed for the new optimization. We also study optimality conditions of the new optimization and how they relate to the null space properties. Furthermore, the proximal gradient algorithm with extrapolation is proposed to solve the new optimization. It is shown that the objective function satisfies the Kurdyka-Łojasiewicz property, which guarantees that the algorithm converge globally to the locally optimal solutions. Finally, experimental results of randomly generated tensor completion problem and color image inpainting problem demonstrate that the proposed optimization and algorithm outperform some traditional nuclear norm algorithms in CPU time.

This paper introduces a novel minimax regularization based on the difference of epsilon-trace norm and Frobenius norm for multi-dimensional data recovery, and then a new nonconvex and nonsmooth bilevel programming is established through the use of the regularization and the new augmented Lagrangian function, which is applied to image and video restoration. Based on the augmented Lagrange multiplier method and difference-of-convex technique, we design a efficient algorithm and develop a convergence theory by using Kurdyka-Lojasiewicz property. Numerical experiments on some color image and video restorations demonstrate that the proposed approach is usually superior to the compared algorithms in recovery effect and CPU time.

Curvature-based regularization has attracted growing concern in the field of image restoration, benefiting from its favorable geometric properties, such as preserving sharp edges, corners and contrast. Total variation regularization has the ability to promote piecewise smooth property and preserve edges in image processing. Inspired by the advantages of curvature regularization and total variation, in the paper, we first develop a regularization that combines curvature and total variation to explore the geometric characteristics inside high-dimensional data, called total curvature variation (TCV) regularization, which can better preserve local information of the underlying data. We present a new low-rank tensor completion model via TCV and low-rank matrix factorization, which can simultaneously exploits the global low-rank prior and local structure information of data. We solve the proposed minimization problem by using the effective proximal alternating minimization algorithm with guaranteed convergence. Results from experiments on color images, videos, and magnetic resonance images show the superior performance of the proposed method over the compared methods in terms of quantitative and qualitative evaluations.

In this paper, the novel optimization model for solving tensor completion with noise is proposed, its objective function is a convex combination of the minimum nuclear norm and maximum nuclear norm. The necessary condition and sufficient condition of the stationary point and optimal solution are discussed. Based on the proximal gradient algorithm and feasible direction method, we design the new algorithm for solving the proposed nonconvex and nonsmooth optimization problem and prove that the sub-sequence generated by the new algorithm converges to the stationary point. Finally, experimental results on the random sample completions and images show that the proposed optimization and algorithm are superior to the compared algorithms in CPU time or precision.

In this paper, an efficient algorithm is proposed for Toeplitz matrix recovery via hybrid thresholding operator. The algorithm is based on the mean-value augmented Lagrangian multiplier algorithm and the singular values are processed by hybrid singular value threshold operator. The new algorithm ensures that the matrix generated by the iteration has a Toeplitz structure, which reduces the calculation time and obtains a more accurate Toeplitz matrix. The convergence of the new algorithm is discussed under certain assumptions. Numerical experiments show that the new algorithm achieves lower CPU time than the mean-value augmented Lagrangian multiplier algorithm, smooth augmented Lagrangian multiplier algorithm, and augmented Lagrangian multiplier algorithm.

This paper studies sparse nonlinear least squares problems, where the Jacobian matrices are unavailable or expensive to compute, yet have some underlying sparse structures. We construct the Jacobian models by the ℓ _1 minimization subject to a small number of interpolation constraints with interpolation points generated from some certain distributions, and propose a derivative-free Levenberg–Marquardt algorithm based on such Jacobian models. It is proved that the Jacobian models are probabilistically first-order accurate and the algorithm converges globally almost surely. Numerical experiments indicate the efficiency of the proposed algorithm for sparse nonlinear least squares problems.

In recent years, numerous studies have proposed leveraging the low-rank structure of high-dimensional data by minimizing the Tensor Ring Nuclear Norm (TRNN), a convex approximation of the Tensor Ring (TR) rank. However, conventional TRNN assigns equal weight to all singular values of the unfolding matrices, disregarding their distinct roles in capturing the physical meanings of the data. To address this limitation, we introduce a novel non-convex smooth approximation for the TR rank based on the Laplace function. This method integrates the Laplace function into TRNN to preserve larger singular values while aggressively penalizing smaller ones in the unfolding matrices. We solve the resulting optimization problem using the Alternating Direction Method of Multipliers (ADMM) algorithm, providing theoretical convergence guarantees. Comprehensive evaluations conducted on diverse datasets, including color images, multispectral images, and color videos, validate that our method achieves superior performance compared to existing state-of-the-art techniques.

Based on the minimum and maximum nuclear norm of all mode matrices of a tensor, a bilevel optimization model with an inequality constraint is proposed for tensor completion with noise. By using a special penalty function, a separable unconstrained optimization and the corresponding proximal gradient method are proposed. Furthermore, the objective function is proved to be a Kurdyka-Lojasiewicz (K L) function with an exponent of 12 , which guarantees that the sequence produced by the proposed algorithm globally converges to a stationary point. Moreover, numerical experiments on random tensor completion and real image recovery show that the proposed model and algorithm are superior to some traditional nuclear models and algorithms in terms of CPU time or precision.

In this paper, we establish a non-convex L & lowast;-LF model for low Tucker rank tensor completion problem. For the new optimization model, three algorithms for solving tensor completion are designed based on the proximal difference of convex algorithm with extrapolation. In theory, the null space property and restricted isometric prop-erty condition are discussed and the new bound of restricted isometry constant delta 2ris given. The optimization objective function is proved to be Kurdyka-& Lstrok;ojasiewicz(KL) function with exponent12. Convergence theory of these algorithms is established, which globally converges to the point of the first-order optimality conditions underKL property. Furthermore, numerical experiments are implemented by these proposed algorithms for the new optimization model and the corresponding algorithms for other models on simulation data and real data. Experimental results show the new models outperform the nuclear norm model and non-convex Schattenp-norm model in pre-cision and CPU times.

. We propose a novel tensor completion model that approximates the tensor rank through a convex combination of the maximum L & lowast; - LF and minimum L & lowast; - LF mode-matrix ranks, thereby reducing the computational complexity associated with mode-matrix operations. To efficiently solve this model, we design a proximal difference-of-convex algorithm with extrapolation. We further examine the theoretical properties of the proposed model, including the null space property and the restricted isometry property. By analyzing the Kurdyka- Lojasiewicz property of the potential function, we ultimately establish the convergence of the proposed method. Extensive numerical experiments on random tensor completion and real-world image recovery illustrate that the proposed model achieves comparable or superior recovery accuracy while significantly reducing CPU computation time compared to existing methods.

In this paper, we propose a novel proximal point Lagrangian-based method for solving convex programming problems with linear equality constraints, where the proximal centers are constructed using convex combinations of the iterates. The new method preserves all the favorable characteristics of customized proximal point algorithm, including convergence of both the primal and dual iterates, as well as the ability to derive closed-form solutions for subproblems under certain conditions. Furthermore, we prove the global convergence and establish an O(1/K) ergodic sublinear convergence rate of our algorithm under mild assumptions. Finally, numerical experiments conducted on basis pursuit and equality-constrained quadratic programming problems demonstrate the superior performance of our proposed algorithm.

In this paper, we establish a non-convex L_*-L_F model for low Tucker rank tensor completion problem. For the new optimization model, three algorithms for solving tensor completion are designed based on the proximal difference of convex algorithm with extrapolation. In theory, the null space property and restricted isometric property condition are discussed and the new bound of restricted isometry constant δ _2r is given. The optimization objective function is proved to be Kurdyka–Łojasiewicz (KL) function with exponent 1/2 . Convergence theory of these algorithms is established, which globally converges to the point of the first-order optimality conditions under KL property. Furthermore, numerical experiments are implemented by these proposed algorithms for the new optimization model and the corresponding algorithms for other models on simulation data and real data. Experimental results show the new models outperform the nuclear norm model and non-convex Schatten p-norm model in precision and CPU times.

Progressive hedging algorithm (PHA) is a long-standing algorithm originally designed for stochastic programming problems and has recently been extended to solving multistage stochastic variational inequalities (SVIs), both are important tools for processing mathematical programming problems with uncertainty. In this note, we first show that PHA for two-stage stochastic linear complementarity problems, a special case of multistage SVIs, is an application of the Douglas-Rachford operator splitting algorithm and then extend this result to multistage SVIs. To do this, we first reformulate the focused problems as zero-finding problems of the sum of two maximal monotone operators. This reformulation plays a key role in establishing the connections Since Douglas-Rachford operator splitting algorithm can be viewed as a proximal point algorithm, our result sharpens the understanding of PHA.

In the past few years, tensor robust principal component analysis (TRPCA) which is based on tensor singular value decomposition (t-SVD) has got a lot of attention in recovering low-rank tensor corrupted by sparse noise. However, most TRPCA methods only consider the global structure of the image, ignoring the local details and sharp edge information of the image, resulting in the unsatisfactory restoration results. In this paper, to fully preserve the local details and edge information of the image, we propose a new TRPCA method by introducing a total generalized variation (TGV) regularization. The proposed method can simultaneously explore the global and local prior information of high-dimensional data. Specifically, the tensor nuclear norm (TNN) is employed to develop the global structure feature. Moreover, we introduce the TGV, a higher-order generalization of total variation (TV), to preserve the local details and edges of the underlying image. Subsequently, the alternating direction method of multiplier (ADMM) algorithm is introduced to solve the proposed model. Sufficient experiments on color images and videos have demonstrated that our method is superior to other comparison methods.

To solve the tensor completion problem,a cyclic algorithm for low rank tensor completion is proposed.Based on the alternating direction multiplier method,the sub-problem is circularly updated,which effectively reduces the cost of tensor expansion,matrix folding and singular value decomposition in the iterative process.At the same time,the convergence analysis of the algorithm is given under reasonable assumptions.Finally,the numerical experiments show that the proposed algorithm is more efficient than other algorithm.

With wide-spread real-world applications, low-rank and sparse matrix recovery, where the concerned matrix with incomplete data is divided into a low-rank part and a sparse part, recently has attracted significant interest. To solve this structured non-convex optimization problem, we propose a non-monotone alternating Newton-like directional method which essentially updates two blocks of variables associated with the low-rank part using a single step of simple line-search along the Newton-like descent directions and another block of variables associated with the sparse part using a non-monotone search. In particular, the non-monotone search technique helps our method find a better Newton-like descent direction in the next step. Moreover, we prove the global convergence of the proposed algorithm and discuss the iteration number in given precision under some mild conditions. Finally, computational results show the efficiency of the developed algorithm.