Wei Li

In this paper, we extend the definition of tensor products from using invertible transformations to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. On the basis of this new definition, we define the concepts of tensor rank and tensor nuclear norm and investigate their properties, ensuring consistency with their matrix counterparts. We then derive a singular value thresholding (SVT) formula to approximately solve the subproblems in the alternating direction method of multipliers (ADMM), which is a key component of our proposed tensor robust principal component analysis (TRPCA) algorithm. We conduct a complexity analysis of the proposed algorithm, demonstrating its computational efficiency, and apply it to grayscale video denoising and motion detection problems, where it shows significant improvements in efficiency while maintaining a similar level of quality. This work provides a promising approach for handling large-scale data, offering new insights and solutions for advanced data analysis tasks.

In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of Carathéodory weak solutions for differential set-valued inverse variational inequalities. Then, by the existence result, we establish the stability for the differential set-valued inverse variational inequality problem when the constraint set and the mapping are perturbed by two different parameters. The closedness and continuity of Carathéodory weak solutions with respect to the two different parameters are obtained.

We consider the partial differential set-valued variational inequality (PDSVI) on real Banach spaces. We give conditions for the existence and uniqueness of mild solution of the PDSVI, and establish closedness and continuity of the mild solution set mapping of the PDSVI. The proofs of the main results are based on the theory of semigroups, set-valued mappings, and variational inequality.

In this paper, we are denoted to introducing the strict feasibility of a variational inclusion problem as a novel notion. After proving a new equivalent characterization for the nonemptiness and boundedness of the solution set for the variational inclusion problem under consideration, it is proved that the nonemptiness and boundedness of the solution set for the variational inclusion problem with a maximal monotone mapping is equivalent to its strict feasibility in reflexive Banach spaces.

In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Caratheodory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.