Properties Of The Solution Set For The Complementarity Problem Related To Tensors

With the development of technology,people are facing many problems related to tensors.In this paper,we study complementarity problems related to tensors,including tensor equations,tensor complementarity problems,generalized polynomial complemen-tarity problems and tensor eigenvalue complementarity problems.Main results are as follows:1.For tensor equation problems,we study properties of the nonnegative solution set.For the tensor equation with triangular tensors,we obtain the existence and uniqueness of nonnegative（positive）solutions under certain cases.For the tensor equation with B-tensors,B₀-tensors,row diagonally dominant tensors or row strictly diagonally dominant tensors,we obtain the nonexistence of nonnegative solutions.For the tensor equation defined by nonnegative tensors with all diagonal entries being positive,B-tensors or row strictly diagonally dominant tensors,we obtain upper bound estimations of nonnegative solution sets.2.For tensor complementarity problems,we study special tensors and the boundary estimation of the solution set of the tensor complementarity problem.For the generalized row strictly diagonally dominant tensor and the tensor complementarity problem with this class of tensors,we first study the relationship among generalized row strictly diagonally dominant tensors and several special tensors.We next study boundary estimation of the solution set of the tensor complementarity problem with generalized row strictly diagonal-ly dominant tensors,and obtain the boundary formula which is easy to calculate.For the tensor complementarity problem which has a solution,we study the lower bound of the solution set,and obtain a lower bound formula which is easy to calculate.For R₀-tensors and tensor complementarity problems defined by this class of tensors,we study properties of R₀-tensors,and obtain an equivalent condition via a new given quantity.Next,we study the upper bound estimation of the tensor complementarity problem with R₀-tensors and obtain an upper bound formula.3.For generalized polynomial complementarity problems,we study the error bound of the solution set.For solvable generalized polynomial complementarity problems,by two different residual functions and using two methods to substitute variables,we obtain two error bounds.Compared with the corresponding known results,our results improve the exponent of the residual function.4.For tensor eigenvalue complementarity problems,we study properties of Pareto eigenvalues defined by the tensor eigenvalue complementarity problem,and mainly study inclusion intervals of Pareto eigenvalues.First,we study basic properties of tensor Pareto eigenvalues,we obtain an equivalent condition of existence for the strict Pareto eigenval-ue,nonnegativity（positivity）of Pareto eigenvalues of（strictly）semi-positive tensors,and an equivalent condition of（strict）copositivity for tensors.Next,we study the inclusion interval of tensor Pareto eigenvalues,three main inclusion intervals are obtained.Our results are tighter than the existing one.At the same time,while considering the inclusion interval of matrix Pareto eigenvalues,our results are tighter than a recent result under cer-tain conditions.As an application of inclusion intervals,we propose a sufficient condition for judging strict copositivity of a tensor.