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Linear complementarity problem over tensor spaces

With the aid of a contraction product of tensors, we define a class of linear complementarity problems over tensor spaces, which is abbreviated as the tensor linear complementarity problem. When the tensor variable involved is reduced to a vector, the considered problem reduces to the classical linear complementarity problem. In this paper, we discuss certain properties of the contraction product of tensors, and study the theory and algorithm of the concerned problem. Specifically, we first discuss an equivalent model of the problem, the feasibility and solvability of the problem, and convexity of the solution set. Then we propose an extragradient method for solving the tensor linear complementarity problem, and show the convergence of the method under suitable assumptions. The preliminary numerical results are also reported.