The Sparsest Solutions to $Z$-Tensor Complementarity Problems

Abstract

Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved ${\ell }_0$ 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.

References in corpus (2)

• RSP-Based Analysis for Sparsest and Least ${\ell }_1$-Norm Solutions to Underdetermined Linear Systems
• Positive Definite Tensors to Nonlinear Complementarity Problems

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