Solvability of Two Classes of Tensor Complementarity Problems

Xu, Yang; Gu, Weizhe; Huang, He

doi:https://doi.org/10.1155/2019/6107517

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2019 | Article ID 6107517 | https://doi.org/10.1155/2019/6107517

Solvability of Two Classes of Tensor Complementarity Problems

Yang Xu,¹Weizhe Gu,¹and He Huang²

Academic Editor: Vyacheslav Kalashnikov

Received18 Oct 2018

Revised23 Jan 2019

Accepted12 Feb 2019

Published04 Mar 2019

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 a -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.

1. Introduction

Denote and where is the set of real numbers. For any set , we use to denote the linear space generated by ; and and to denote the boundary and the interior of , respectively. A set is said to be convex if for any and any , it follows that ; and it is said to be a cone if for any and any , it follows that . A cone is closed convex if it is a closed and convex set. For any set , we use to denote the dual cone of , which is defined by

An th order -dimensional real tensor is a multidimensional array with real entries, in which for . We denote the set of all the th order -dimensional real tensors by . For any and , an -dimensional vector, denoted by , is defined by

For any given and , the tensor complementarity problem [1], denoted by the TCP(, ), is to find an such thatwhere means for all . Obviously, the TCP(, ) is a generalization of the linear complementarity problem [2]. Recall that, for any given closed convex cone and mapping , the conic complementarity problem [3], denoted by the CP(, ), is to find a vector such thatwhere is the dual cone of . It is obvious that the TCP(, ) is a subclass of the CP(, ).

As a natural extension of the linear complementarity problem, the TCP(, ) emerged from the tensor community in 2015 [1]. In recent several years, the TCP(, ) has been a hot topic and many theoretical results have been obtained, including the nonemptiness and/or compactness of solution set [4–16], the existence of unique solution [7, 8, 14, 17–22], error bound theory [23–25], strict feasibility [22, 26], and so on. Several algorithms for solving the TCP(, ) have also been proposed [27–31]. Recently, Wang, Hu, and Huang [14] investigated the quadratic complementarity problem, i.e., find an such thatwhere , , and , which serves as an important bridge linking linear complementarity problems and nonlinear complementarity problems. They studied various properties on the solution set for a quadratic complementarity problem, including existence, compactness, and uniqueness.

In this paper, we first introduce a class of structure tensors, called the positive semidefinite plus tensor, which is a generalization of the positive semidefinite plus matrix used in [14]; and we discuss its simple properties. Then, we investigate two classes of tensor complementarity problems, which contain the quadratic complementarity problem (5) as a special case. In particular, 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.

It has been shown that a multiperson noncooperative game can be reformulated as a tensor complementarity problem, and that finding a Nash equilibrium point of the multiperson noncooperative game is equivalent to finding a solution of the resulting tensor complementarity problem [32]. It is well known that many engineering problems can be solved by solving some clustering problems. In the age of big data, hypergraph clustering has become a research hotspot. Recently, game-theoretic approaches for hypergraph clustering have been developed rapidly [33]. 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. In order to further expand the scope of engineering application of this kind of problem, in this paper we investigate a static traffic equilibrium problem, which can be reformulated as a concerned complementarity problem. A specific example is also given.

The remainder of the paper is organized as follows. In the next section, we introduce the positive semidefinite plus tensor and discuss its simple properties. In Section 3, we discuss properties of solution sets of two classes of tensor complementarity problems. In Section 4, we reformulate a static traffic equilibrium problem as a concerned complementarity problem. Conclusions are given in the last section.

Throughout this paper, for any and , we will use the following notation:

2. A New Class of Tensors and Its Properties

The following tensors are defined similarly as those given in [34], which will be used in this paper.

Definition 1. Let be a closed cone. A tensor is said to be (a)positive semidefinite on if for all ;(b)positive definite on if for all ;(c)strictly positive semidefinite on if for all ;(d)strictly positive definite on if for all with .

Inspired by Definition 3.3 in [14], we introduce a new class of structured tensors as follows.

Definition 2. Let be a closed cone. is said to be positive semidefinite plus on if it is positive semidefinite on , and whenever for , we have .

Recall that if there exist a scalar and a nonzero vector which satisfythen is called an -eigenvalue of with being corresponding -eigenvector; and that if there exist a scalar and a nonzero vector which satisfythen is called a -eigenvalue of with being corresponding -eigenvector [35]. By these definitions and Definition 2, the following result is obvious.

Proposition 3. Let be a closed cone. If is positive semidefinite plus on , then for any satisfying , it follows that is an -eigenvalue of with being corresponding -eigenvector; and is a -eigenvalue of with being corresponding -eigenvector.

The following two propositions further give some simple properties of the positive semidefinite plus tensor.

Proposition 4. Let be positive semidefinite plus on . If there exists such that , then for all .

Proof. Let denote the vector whose th entry is and others are . Suppose that there exists such that , then we have which contradict that is positive semidefinite plus on . Thus, the desired result holds.

Proposition 5. Let be a closed cone. is positive semidefinite plus on if and only if either (a) is positive definite on , or(b) is positive semidefinite on ; and for any , is an -eigenvector (-eigenvector) of corresponding to -eigenvalue (-eigenvalue) .

Proof. By Definition 2 and Proposition 3, it is easy to show that the desired result holds.

3. Properties of Solution Sets

For any given , we use . Given a point , is the cone and where means the dual cone of the set .

Definition 6 (see [14]). Let be a closed cone. A tensor is called -strictly copositive if is copositive and for all .

Lemma 7 (see [3]). Let be a closed convex cone and be continuous. Consider the following statements: (a)There exists a vector such that the set is bounded (possibly empty).(b)There exists a bounded open set and a vector such that (c)The CP(, ) given by (4) has a solution. It holds that (a)(b)(c). Moreover, if the set , which is nonempty and larger than , is bounded, then CP(, ) given by (4) has a nonempty and compact solution set.

The following theorem is an extension of Proposition 3.3 in [14].

Theorem 8. Let be -strictly copositive on a closed cone , be positive semidefinite plus on a closed cone , and be a given vector. Let be the intersection of the solution set of and the subspace generated by . Suppose that , andThen, the generalized tensor complementarity problem has a nonempty and bounded solution set.

Proof. The proof of this theorem depends heavily on the idea given in the proof of [14, Proposition 3.3]. For completeness, we give it as follows.
By Lemma 7, we only need to show that the set is bounded. We assume that is an unbounded sequence and derive a contradiction by the method of induction.
Since the sequence is bounded, there exists a subsequence such that and . Furthermore, taking subsequence of if necessary, we assume that either or . Since is closed and , we have that if .
On one hand, it is easy to see thatThus, by dividing the inequality by and letting , we obtain that because is copositive. Since is strictly copositive on , we further haveSo, we obtain that and .
On the other hand, by (16) and the copositivity of we have thatThus, by dividing the inequality by and letting , it follows from (18) that since . Furthermore, we conclude that since is positive semidefinite plus on . So, we have that , i.e.,Furthermore, since and is positive semidefinite plus on , it follows from (18) that , which implies that . Clearly, this, together with (17) and (19), contradicts (13).
The proof is complete.

Corollary 9. Let be -strictly copositive on a closed cone , be positive semidefinite plus on a closed cone for some , be copositive on for all with and , and be a given vector. Let be the interaction of the solution set of and the subspace generated by . Suppose that , and Then, the tensor complementarity problem has a nonempty and bounded solution set.

Proof. Clearly, we have because is copositive on . In the proof of Theorem 8, we know that the set is bounded. Then, we have that the set is bounded. Furthermore, the desired result holds by Lemma 7.

Theorem 10. Let and for some be positive semidefinite on a closed convex cone , and for any . Iffor , then the CP(, ) has a solution.

Proof. Denote . We consider the following three optimization problems:andwhere . It is obvious that problems (24), (25), and (26) are solvable because objective functions , , and are continuous and the feasible set is nonempty and compact. Furthermore, it follows that the optimal solution sets of three optimization problems are compact by the continuity argument. We divide the proof into the following two parts.
Part 1. Suppose that . In this case, for any and , we have that which implies that is bounded (possibly empty). Thus, it follows from Lemma 7 that the CP(, F) has a solution.
Part 2. Suppose that , i.e., for all . In this case, for any , it follows from (23) that and , which implies that and . Denote Then, it follows that . For any , we have Thus, for all and , which implies that is bounded. So, it follows from Lemma 7 that CP(, F) has a solution.
Combining Part 1 with Part 2, we complete the proof.

Lemma 11 (see [3]). Let be a closed convex cone, and be continuous. (a)If is strictly monotone on , i.e., for any , with , then the CP(, ) has at most one solution.(b)If is strongly monotone on , i.e., there exists a constant such that for any , , then the CP(, ) has a unique solution.

Theorem 12. Let and for some be strictly positive semidefinite on a closed convex cone , and for. If for all with , then the CP(, ) has a unique solution.

Proof. By assumptions it follows that, for any with , which implies that the function is strictly monotone on , and hence, the CP(, ) has at most one solution by Lemma 11.
On the other hand, since and are strictly positive semidefinite on and , it follows that and are positive semidefinite on ; and the assumption that for any with implies that for all .
Therefore, by Theorem 10, the desired result holds.

Remark 13. In Theorems 10 and 12, when , we can obtain that the function is not strongly monotone on , since is a cone and for all . Thus, Theorem 12 gives a new existence result of unique solution to the concerned complementarity problem.

4. Traffic Equilibrium Problems

In this section, we describe a static traffic equilibrium problem [3] as an example for applications of the complementarity problems concerned in this paper. We use the same notations as in [3]. Suppose that there is a network given by a set of nodes and a set of arcs . Assume that users compete noncooperatively for the network and attempt to minimize their costs, where the cost of traveling along an arc is a nonlinear function of the total flow vector with components for all . Let be the vector with components for all . We denote the set of origin nodes by and the set of destination nodes by , and they are distinguished. Further, the set of origin-destination (OD) pairs is a given subset of . For each , let denote the set of paths connecting the OD pair , and let be the union of ranging over all . Let denote the flow on the path , and let be the travel cost of flow on this path which is a function of the vector of path flows. Let be the arc-path incidence matrix with entries Obviously, and satisfy the relationship: . A common assumption on the path cost function is that it satisfies The unknown vector where represents the minimum travel cost between OD pair . Let with , where denote the given fixed travel demand between OD pair . The Wardrop equilibrium principle [36] states that each user will choose the minimum cost path between every OD pair, and through this process the paths that are used will have equal cost; paths with costs higher than the minimum will have no flow. Thus,Moreover, the travel demand must be satisfied, i.e.,and the minimum travel cost must be nonnegative, i.e.,Under some assumptions (see, for example, [3, Proposition 1.4.6]), conditions (37)-(39) are equivalent the following complementarity problem: where is the (OD pair, path)-incidence matrix whose entries are Specially, we assume that , , and is a polynomial mapping of degree 2, i.e., , where is a -order -dimensional tensor, is an matrix, and . Then the above complementarity problem becomeswhere Obviously, problem (42) is a specific case of the complementarity problems concerned in Section 3.

In the following, we give a specific example for this class of traffic equilibrium problems, which has only an OD pair for simplicity.

Example 1. We consider a network given by a set of nodes , where the origin node is and the destination node is . There are four arcs: , , , and . We denote , , , and . There are two paths: and , connecting the origin and the destination , where the path traverses arcs and , and the path traverses arcs and . The cost of travel along , , , and is defined as follows: where is the total flow vector. The travel demand is constant . The minimum travel cost is denoted by . The arc-path incidence matrix is Let and denote the flow on the paths and , respectively. Thus, we have and Clearly, , , and satisfy the conditions of [3, Proposition 1.4.6]. Thus, the static traffic equilibrium problem is equivalent to the following complementarity problem: which can be rewritten as where It is clear that is a solution of this complementarity problem.

5. Conclusions

In this paper, we introduced a new class of tensors and discussed its simple properties. We mainly studied properties of solution sets of two classes of tensor complementarity problems, including the nonemptiness and compactness of solution set and existence of unique solution. We also investigated an application of the concerned complementarity problem in terms of a static traffic equilibrium problem. One of further issues is to design effective algorithms to solve the concerned complementarity problem; and another further issue is to investigate the error bound theory for the complementarity problems concerned in this paper.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grants nos. 11431002 and 11871051).

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Copyright © 2019 Yang Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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