Tensor Complementarity Problems—Part I: Basic Theory

• Published in: Journal of optimization theory and applications Vol. 183; no. 1; pp. 1 - 23
• Main Authors: Huang, Zheng-Hai, Qi, Liqun
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Error analysis; Invited Paper; Mathematical analysis; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Stability analysis; State-of-the-art reviews; Tensors; Theory; Theory of Computation; Error bound theory; 90C30; 90C33; Tensor complementarity problem; Stability and continuity analysis; 65H10; Global uniqueness and solvability; Nonemptiness and compactness of the solution set
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-019-01566-z

Tensors (hypermatrices) are multidimensional analogs of matrices. The tensor complementarity problem is a class of nonlinear complementarity problems with the involved function being defined by a tensor, which is also a direct and natural extension of the linear complementarity problem. In the last few years, the tensor complementarity problem has attracted a lot of attention, and has been studied extensively, from theory to solution methods and applications. This work, with its three parts, aims at contributing to review the state-of-the-art of studies for the tensor complementarity problem and related models. In this part, we describe the theoretical developments for the tensor complementarity problem and related models, including the nonemptiness and compactness of the solution set, global uniqueness and solvability, error bound theory, stability and continuity analysis, and so on. The developments of solution methods and applications for the tensor complementarity problem are given in the second part and the third part, respectively. Some further issues are proposed in all the parts.