Abstract
In this paper, we first introduce a class of tensors, called positive semidefinite plus tensors on a closed cone, and discuss its simple properties; and then, we focus on investigating properties of solution sets of two classes of tensor complementarity problems. We study the solvability of a generalized tensor complementarity problem with aD-strictly copositive tensor and a positive semidefinite plus tensor on a closed cone and show that the solution set of such a complementarity problem is bounded. Moreover, we prove that a related conic tensor complementarity problem is solvable if the involved tensor is positive semidefinite on a closed convex cone and is uniquely solvable if the involved tensor is strictly positive semidefinite on a closed convex cone. As an application, we also investigate a static traffic equilibrium problem which is reformulated as a concerned complementarity problem. A specific example is also given.
Highlights
IntroductionFor any given A ∈ Tm,n and q ∈ Rn, the tensor complementarity problem [1], denoted by the TCP(q, A), is to find an x ∈ Rn such that x ≥ 0, Axm−1 + q ≥ 0,
Denote [n] fl {1, 2, . . . , n} and Rn fl {x = (x1, x2, . . . , xn)⊤ : xi ∈ set S R, i ∈ ⊆ Rn,[n]} where R is the set of we use lin(S) to denote the real numbers
We first introduce a class of tensors, called positive semidefinite plus tensors on a closed cone, and discuss its simple properties; and we focus on investigating properties of solution sets of two classes of tensor complementarity problems
Summary
For any given A ∈ Tm,n and q ∈ Rn, the tensor complementarity problem [1], denoted by the TCP(q, A), is to find an x ∈ Rn such that x ≥ 0, Axm−1 + q ≥ 0,. We study the solvability of these two problems and obtain some results which are generalizations of those given in [14] These provide important theoretical basis for designing effective algorithms to solve these two classes of problems. In terms of methods developed in [32], several problems of hypergraph clustering can be reformulated as tensor complementarity problems These provide an impetus for further study of this class of problems. Throughout this paper, for any A ∈ Tm,n and x ∈ Rn, we will use the following notation: n
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