Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved ℓ 0 norm. In this paper, a special type of tensor complementarity problems with Z-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particularly, in comparison to existing exact relaxation conditions, such as RIP-type ones, our proposed conditions are easy to verify.Keywords Z-tensor · tensor complementarity problem · sparse solution · exact relaxation · polynomial programming Mathematics Subject Classification (2000) 90C26 · 90C33 · 15A69 · 53A45

Abstract. It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic hierarchy process and so on. In this paper, the concept of essentially nonnegativity is extended from matrices to higher order tensors, and the convexity and log convexity of dominant eigenvalues for such a class of tensors are established. Particularly, for any nonnegative tensor, the spectral radius turns out to be the dominant eigenvalue and hence possesses these convexities. Finally, an algorithm is given to calculate the dominant eigenvalue, and numerical results are reported to show the effectiveness of the proposed algorithm.

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The completely positive (CP) tensor verification and decomposition are essential in tensor analysis and computation due to the wide applications in statistics, computer vision, exploratory multiway data analysis, blind source separation and polynomial optimization. However, it is generally NP-hard as we know from its matrix case. To facilitate the CP tensor verification and decomposition, more properties for the CP tensor are further studied, and a great variety of its easily checkable subclasses such as the positive Cauchy tensors, the symmetric Pascal tensors, the Lehmer tensors, the power mean tensors, and all of their nonnegative fractional Hadamard powers and Hadamard products, are exploited in this paper. Particularly, a so-called CP-Vandermonde decomposition for positive Cauchy-Hankel tensors is established and a numerical algorithm is proposed to obtain such a special type of CP decomposition. The doubly nonnegative (DNN) matrix is generalized to higher order tensors as well. Based on the DNN tensors, a series of tractable outer approximations are characterized to approximate the CP tensor cone, which serve as potential useful surrogates in the corresponding CP tensor cone programming arising from polynomial programming problems.

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