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On the properties of tensor complementarity problems

Properties of solutions of the tensor complementarity problem (TCP) for structured tensors have been investigated in recent literature. In this paper, we make further contributions on this problem. Specifically, we first derive solution existence theorems for TCPs on general cones from the results studied in the nonlinear complementarity problem literature. An interesting byproduct is that conditions (e.g., strict copositivity) of solution existence results for TCPs on the nonnegative cone can be reduced to copositivity, which, to the best of our knowledge, is the weakest requirement in the current TCP literature. Moreover, we study the topological properties of the solution set and stability of the TCP at a given solution, which are not discussed before and further enrich the theory of TCPs.


Introduction
The complementarity problem is a long historical topic that has become a wellestablished and fruitful discipline within mathematical programming. We here refer the reader to monographs [4,7,14] and surveys [8,11] for the well developed basic theory, numerical algorithm, and applications of complementarity problems.
In recent decades, with the rapid development of the discipline of tensors, the so-called tensor complementarity problem (TCP) over the nonnegative cone was introduced recently for the research on structured tensors [19]. It is well known that the concept of tensors is a natural generalization of matrices. Therefore, the classical linear complementarity problem (LCP) is a special case of TCPs. However, most of the well established properties of LCPs cannot be extended to TCPs directly due to the complicated structure of tensors. According to the definition of TCP, we can easily see that such a model falls into a special case of the nonlinear complementarity problem (NCP), and thus many results of NCPs are certainly applicable to TCPs, which, however, do not often embody the structure of tensors.
Comparatively, we are more interested in some specialized properties of TCPs by fully considering the structure of tensors. For instance, some recent papers are dedicated to showing the existence of solutions of TCPs with some structured tensors (e.g., nonnegative tensors [13,19], symmetric positive definite tensors and copositive tensors [3], (strictly-) semi-positive tensors [20,22], Z-tensors [10] and M-tensors [5,23], ER-tensors [1], and P-tensors [1,6,19]). In [1], the authors considered the property of global uniqueness and solvability for TCPs with particular tensors. Along with the booming of sparse optimization, Luo et al. [16] studied the sparsest solutions to TCPs. Most recently, Huang and Qi [12] reformulated an n-person noncooperative game as a TCP showing an interesting application of TCPs in management science.
Note that all results mentioned above focus on the special case, TCPs over the nonnegative cone, and require some relatively stronger conditions. Therefore, there are three natural questions: i) can we consider the TCP on a general cone?
ii) can we establish the solution existence theorem under weaker conditions on tensors than the previous results? iii) what more properties can we obtain for the problem under consideration?
In this paper, we consider the TCP over a general cone extending the model introduced in [19]. As an special case of NCPs, we derive some specific solution existence results for the TCPs under consideration from the results presented by Gowda and Pang [9]. An interesting consequence is that conditions (e.g., strict copositivity) of the solution existence for TCPs over the nonnegative cone can be weakened to copositivity, which is weaker than the requirements investigated in the TCP literature. The other two important results are the topological properties of the solution set and stability of solutions of the general TCP. To the best of our knowledge, such properties are not studied in the current TCP literature. All results presented in this paper further enrich the theory of TCPs.
The structure of this paper is organized as follows. In Section 2, we first describe the general model of the TCP under consideration and introduce some notations that will be used throughout. In Section 3, we summarize some definitions and properties, which are preparations of the subsequent analysis. In Section 4, based on the results of Gowda and Pang [9], we present the existence of solutions of TCPs under mild conditions. In Section 5, we show the topological properties and stability of solutions of general TCPs. Finally, we complete this paper with drawing some concluding remarks in Section 6.

The Model and Notation
The concept of tensors is a natural generalization of matrices. Notationally, let A := (a i1i2...im ) 1≤i1,i2,...,im≤n denote an m-th order n-dimensional square tensor, where a i1i2...im ∈ R. Denote by Tm,n the space of all m-th order n-dimensional square tensors. Clearly, we can see that Tm,n is a linear space of dimension n m .
Here, we shall mention that all the tensors discussed in this paper are real. Besides, we denote by I := (σ i1···im ) the unit tensor in Tm,n, where σ i1···im is the Kronecker For given A ∈ Tm,n and q ∈ R n , the tensor complementarity problem (TCP) refers to the task of finding a vector x ∈ R n such that x ∈ K, w := Ax m−1 + q ∈ K * , and x ⊥ w := x, w = 0, (2.1) where K ⊂ R n is a given closed and convex pointed cone, K * is the dual cone of K defined by and ·, · denotes the standard inner product in real Euclidean space, the function F (x) := Ax m−1 : R n → R n whose i-th component is given by Throughout, we define Ax m as the value at x of a homogeneous polynomial of the form Obviously, each component of Ax m−1 is a homogeneous polynomial of degree m − 1. Note that TCP (2.1) immediately reduces to the model studied in [1,19,20] when taking K := R n + . In what follows, we denote TCP (2.1) by TCP(K, q, A) for notational convenience. Moreover, we denote by SOL(K, q, A) the solution set of TCP(K, q, A), i.e., It is noteworthy that the solution set SOL(K, q, A) is possibly empty for general tensors. Therefore, it is necessary to investigate that what tensors could make the solution set SOL(K, q, A) nonempty and what more properties we can get for such tensors.
Throughout this paper, let In := {1, 2, · · · , n} be an index set. Denote R n the real Euclidean space of column vectors with length n, i.e., where R is the set of real numbers and the symbol ⊤ represents the transpose.
Let en := (1, 1, . . . , 1) ⊤ with length n. Let Bn(x, r) represent the closed ball centered at x with radius r in R n , and in particular, denote by Bn the unit sphere centered at 0 ∈ R n . For a given subset N in R n , denote by cl(N ) the topological closure of N .
For given A ∈ Tm,n and a nonempty subset α of In, we denote the principal sub-tensor of A by Aα, which is obtained by homogeneous polynomial Ax m for all x := (x 1 , x 2 , . . . , xn) ⊤ with x i = 0 forᾱ = In\α. So, Aα ∈ T m,|α| , where the symbol |α| denotes the cardinality of α. Correspondingly, we denote the sub-tensor of A by Aᾱα, which consists of the elements a i1i2...im in A with i 1 ∈ α and i 2 , . . . , im ∈ α.
Notice that Aᾱα is not a square tensor. If the entries a i1i2...im of a given tensor A are invariant under any permutation of their indices, then A is called a symmetric tensor. If for every i ∈ In, A i := (a ii2...im ) 1≤i2,...,im≤n , an (m − 1)-th order ndimensional square tensor, is symmetric, then A is called a sub-symmetric tensor with respect to the indices {i 2 , . . . , im}. Apparently, a symmetric tensor A must be sub-symmetric, but the reverse is not true. For given A := (a i1i2...im ) ∈ Tm,n and x ∈ R n , Ax m−2 denotes the n × n matrix with its (i, j)-th element given by

Definitions and Lemmas
In this section, we introduce some basic definitions and lemmas, which pave the way of our further analysis.
Let K be a closed and convex pointed cone in R n . Recall that a nonempty set Ω ⊂ R n generates K, thereby writing K := cone(Ω) if K := {ts : s ∈ Ω, t ∈ R + }. If in addition Ω does not contain the zero vector and for each x ∈ K\{0}, there exist unique s ∈ Ω and t ∈ R + such that x = ts, then we say that Ω is a basis of K. Whenever Ω is a finite set, cone(conv(Ω)) is called a polyhedral cone, where conv(Ω) stands for the convex hull of Ω. It is clear that R n + is a closed convex cone in R n , whose a compact basis is ∆ : Definition 3.1 Let A = (a i1i2...im ) ∈ Tm,n, and let K be a given closed convex cone in R n . Then, A is said to be (i) K-positive semi-definite, if Ax m ≥ 0 for any vector x ∈ K; (ii) K-positive definite, if Ax m > 0 for any vector x ∈ K\{0}.
The following property characterizes K-positive semi-definite (definite) tensors and extends ones proposed in [17]. For the sake of completeness, here we still present its proof. Proof For every x ∈ K\{0}, there exist unique s ∈ Ω and t ∈ R ++ such that x = ts. Consequently, the desired result follows from Definition 3.1. ⊓ ⊔ Definition 3.2 Let A ∈ Tm,n, and let K be a given closed convex pointed cone in R n . We say that A is K-regular if it satisfies Definition 3.3 Let A ∈ Tm,n, and let K be a given closed convex cone in R n . We say that A is K-singular if it satisfies Otherwise, we say that A is K-nonsingular. In particular, we say that A is singular Otherwise, A is said to be nonsingular.

From Definition 3.3, it is easy to see that,Ā isK-nonsingular if and only if, for
It is clear that, if A is nonsingular, then for any given closed convex cone K in R n , A is K-nonsingular.
From Definitions 3.1 and 3.3, it is easy to see that, a K-positive definite tensor A must be K-nonsingular. As we know, if a symmetric matrix A ∈ T 2,n is positive semi-definite and nonsingular, then it must be positive definite. However, if A is asymmetric, the above conclusion is not true. The following example shows that, when m ≥ 3, even if A ∈ Tm,n is symmetric, K-positive semi-definiteness and K-nonsingularity of A do not imply the K-positive definiteness of A.
Consequently, it is easy to verify that A is copositive and nonsingular, but not strictly copositive.
Denote by C(R n ) the set of nonzero closed convex cones in R n , which is associated with the natural metric defined by where K 1 , K 2 ∈ C(R n ) and dist(z, K) := inf u∈K z − u stands for the distance from z to K. An equivalent way of defining δ is stands for the Hausdorff distance between the compact sets C 1 , C 2 ⊂ R n (see [2, pp.85-86]). For more details of the metric δ, see [18]. According to Walkup and Wets [21], the operation K → K * is an isometry on the space (C(R n ), δ), that is to say, From Definition 3.3, we immediately obtain the following lemma.
Proof We prove it by contradiction. Suppose that the conclusion is not true, then there exists a sequence of Since {x (l) } ⊆ Bn and Bn is compact, without loss of generality, we assume x (l) → x ∈ Bn as l → ∞. Furthermore, since x (l) ∈ K l ∩ Bn for every l and δ(K l ,K) → 0 as l → ∞, by (3.1), it is not difficult to know thatx ∈K. Accordingly, by letting It is a contradiction, becauseĀ isK-nonsingular andx ∈K\{0}.

⊓ ⊔
Hereafter, for given A ∈ Tm,n and K ∈ C(R n ), we denote Clearly, when taking m = 2 (i.e., A is a matrix), Tpos(R n + , A) reduces to the closed convex cone generated by A (see [4]). However, when m ≥ 3, Tpos(R n is not convex in general, but still remains the closedness that will be proved in the following lemma.
is a closed cone. Proof We first prove that Tpos(K,Ā) is a cone, i.e., ty ∈ Tpos(K,Ā) for any y ∈ Tpos(K,Ā) and t ∈ R + . Since y ∈ Tpos(K,Ā), there exists x ∈K such that y =Āx m−1 , which implies that ty =Ā(t We now prove that Tpos(K,Ā) is closed. Note that the function F defined by (2.2) is a compact map, which implies F (K ∩ Bn) is compact, sinceK ∩ Bn is a compact basis ofK. It is clear that F (K ∩ Bn) ⊆ Tpos(K,Ā), which implies that cone(F (K ∩ Bn)) ⊆ Tpos(K,Ā), since Tpos(K,Ā) is a cone. Moreover, we claim that Tpos(K,Ā) = cone(F (K ∩ Bn)). In fact, for any y ∈ Tpos(K,Ā), there exists x ∈K such that y =Āx m−1 . If x = 0, it is obvious that y = 0 ∈ cone(F (K ∩ Bn)). Without loss of generality, we assume that x ∈K\{0}. Letx := x/ x , and then Notice that theK-nonsingularity ofĀ is only a sufficient condition for the closedness of Tpos(K,Ā), which will be showed in the following example.
In the rest of this section, we regard Tpos(·, ·) as a set-valued map from C(R n )× Tm,n into the power set of R n . The following lemma characterizes the closedness of the map Tpos(·, ·).

⊓ ⊔
Indeed, theK-nonsingularity ofĀ is a key condition for Lemma 3.2. Below, we give an example to show that theK-nonsingularity ofĀ is necessary to ensure the closedness of the map Tpos at (K,Ā), even for a very special case.

Existence of Solutions
The properties of solutions of TCPs have been investigated under certain conditions in earlier papers. In this section, we still study along this line and in particular show more interesting results of the TCP on a nonnegative cone, that is, we prove that TCP(R n + , q, A) has a solution under comparatively weaker conditions. Here, we first make the following assumption. From Lemma 3.1 and Property 3.3, we know that, if Aα is R |α| + -nonsingular for the subset α of In, then Tpos(R n + , C A (α)) is closed. Notice that if A is strictly copositive, then Aα is R |α| + -nonsingular for every nonempty subset α of In. The following example shows that even if A ∈ Tm,n is copositive and Aα is R |α| +nonsingular for every nonempty subset α of In, it is not necessarily strictly copositive.
where a 111 = a 222 = 1, a 112 = a 122 = −1 and others are zeros. Then for any x ∈ R 2 , we have Ax 2 = (x 2 Consequently, it is easy to verify that Aα is nonsingular for every nonempty subset α ⊆ {1, 2} and A is copositive, but not strictly copositive. The above example efficiently shows that the combination of R |α| + -nonsingularity and copositivity is relatively weaker than the strict copositivity. For given K ∈ C(R n ) and A ∈ Tm,n, we now denote (4.1) Proposition 4.1 For given K ∈ C(R n ) and A ∈ Tm,n. Then, Q(K, A) given by (4.1) is a cone. In particular, if A satisfies Assumption 4.1 with K = R n + , then Q(R n + , A) is a closed cone. Proof We first prove that Q(K, A) is a cone, i.e., q ∈ Q(K, A) and t ∈ R + implies tq ∈ Q(K, A). When t = 0 or q = 0, the conclusion is obvious since tq = 0 ∈ Q(K, A). We now assume q = 0 and t > 0. Since q ∈ Q(K, A), we know that SOL(K, q, A) = ∅. Take x ∈ SOL(K, q, A). Then, it is easy to see that t For any q ∈ Tpos(R n + , C A (α)) with some α ⊆ In, there exists u = (uα, uᾱ) ∈ R n + such that q = C A (α)u m−1 , that is, This means that x = (uα, 0) ∈ SOL(R n + , q, A), and hence q ∈ Q(R n + , A). Therefore, we have Conversely, for every q ∈ Q(R n + , A), there exists x ∈ R n such that If x = 0, then q ≥ 0 by (4.3). Consequently, we have q ∈ Tpos(R n + , C A (∅)) since C A (∅) = I by Property 3.3. We consider the case of x = 0. Denote α = {i ∈ In | x i > 0} andᾱ = In\α. It then follows from (4.3) that For such α ⊆ In, we define C A (α) by (3.4) and let u = (xα, (yᾱ) [ 1 m−1 ] ). It is clear that u ∈ R n + . Moreover, it holds that q = C A (α)u m−1 , which implies that there exists a subset α of In satisfying q ∈ Tpos(R n + , C A (α)). Therefore, (4.2) holds. Moreover, for a tensor A ∈ Tm,n, there are 2 n (not necessarily all distinct) Tpos(R n + , C A (α)), which, together with Assumption 4.1 with K = R n + , implies that Q(R n + , A) is closed.
⊓ ⊔ Proposition 4.2 Let q ∈ R n and A ∈ Tm,n. If A satisfies Assumption 4.1 with K = R n + and SOL(R n + , q, A) = ∅, then there exists a neighborhood V of q such that SOL(R n + , q ′ , A) = ∅ for any q ′ ∈ V .
Proof Since SOL(R n + , q, A) = ∅, it holds that q ∈ Q(R n + , A). From Proposition 4.1, we know Q(R n + , A) is closed, which means there exists a neighborhood V of q, such that V ∩ Q(R n + , A) = ∅, and hence the desired result follows.

⊓ ⊔
Let A = (a i1i2...im ) ∈ Tm,n. It is easy to see that SOL(R n + , 0, A) = ∅. If TCP(R n + , q, A) always has a solution for any q ∈ R n , then we call A a Q-tensor. It has been well documented in [1,3,6,10,19,22] that strictly semi-positive tensors, P-tensors, strictly copositive tensors, ER-tensors, positive tensors, and R-tensors are Q-tensors. If A is a strong M-tensor satisfying a i1···im = 0 whenever i j = i k for some j = k, Gowda et al. [10] proved that A is also a Q-tensor. If a symmetric tensor is copositive, then such a tensor is semi-positive. Correspondingly, TCP(R n + , q, A) with a semi-positive A has a unique solution for all q > 0 (see [20]). When the underlying A is a semi-positive R 0 -tensor, the authors of [19] proved that A must be an R-tensor. Consequently, TCP(R n + , q, A) has a solution for any q ∈ R n . In addition, it has been proved in [13] that a nonnegative tensor is also an R-tensor. When we consider a special case of TCP(R n + , q, A) with a second-order tensor (i.e., A is a matrix), it is known from [4] that if A is copositive, then, for all q ∈ R n with the following property v ≥ 0, Av ≥ 0, v ⊤ Av = 0 ⇒ v ⊤ q ≥ 0, TCP(R n + , q, A) (which is actually an LCP) has a solution. Inspired by such a result, we are further concerned with the solution existence of TCPs (m ≥ 3) with copositive tensors.
To simplify the notations, let S A := SOL(R n + , 0, A). It is clear that S A is a cone and S A ⊆ R n + . Consequently, it holds that R n + ⊆ S * A , which implies that int(S * A ) contains R n ++ . For a class of multivalued functions with the "upper limiting homogeneity" (ULH) property, Gowda and Pang [9] derived the existence of a solution to the corresponding multivalued complementarity problems. According to their result, we have the following theorem. Moreover, SOL(R n + , q,Ā) is compact for any q ∈ int(S * Ā ) Proof We first know that Q(R n + ,Ā) is closed, by Proposition 4.1. We now take the multivalued function Φ(x) in [9] as Φ(x) := F (x) :=Āx m−1 , then Φ has the ULH property with degree m−1. Moreover, the function Γ , defined by (3) in [9], is equal to F (x). Consequently, under the condition thatĀ is copositive, by Theorem 2 in [9], we know that SOL(R n + , q,Ā) is a nonempty compact set for every q ∈ int(S * Ā ), which implies int(S * Ā ) ⊂ Q(R n + ,Ā). This implies, together with the closedness of . We obtain the desired results and complete the proof.  (ii) for all x ∈ SOL(R n + , q, A), it hold that x ≤ c.
Proof We first prove part (i) that, there exists a neighborhood N of (q,Ā), such that q ∈ int(S * A ) for any (q, A) ∈ N , by contradiction. Suppose such no neighborhood exist, then there exists a sequences {q (l) , A l , v (l) } ⊂ R n ×Tm,n ×R n satisfying (q (l) , A l ) → (q,Ā) and v (l) ∈ SOL(R n + , 0, A l ), such that (v (l) ) ⊤ q (l) ≤ 0. Without loss of generality, we assume v (l) = 1 for each l. Letṽ be an accumulation point of the sequence {v (l) }. It is easy to see thatṽ is a nonzero solution of TCP(R n + , 0,Ā) and satisfiesṽ ⊤q ≤ 0, which contradicts the assumptionq ∈ int(S * Ā ). Hence, the existence of N follows. Moreover, without loss of generality, we may assume that A satisfies Assumption 4.1 for any (q, A) ∈ N . Consequently, by Theorem 4.1, we know that, for any (q, A) ∈ N with A being copositive, the TCP(R n + , q, A) must have a solution. Hence part (i) is established.
We prove part (ii) by contradiction. Suppose that no such a constant c exists.

Topological Properties and Stability
As far as we know, properties on solution set such as topological properties and stability are discussed much less in TCP literature. Thus, we in this section investigate these results to enrich the theory of TCPs.

Topological properties
We assume no special structure on K other than the fact that it is closed and convex. First, we introduce a concept for the study of topological properties of the solution set. As far as the semicontinuity concepts are concerned, we use the following terminology (see [2, Section 6.2]).
Definition 5.1 Let W and Y be two topological spaces. The mapping Ψ : W → 2 Y is said to be upper-semicontinuous atw ∈ W , if whenever U is an open subset of Y containing Ψ (w), then U contains Ψ (w) for each w in some neighborhood ofw.
Here, we define the solution set of TCPs as the mapping SOL(·, ·, ·) : C(R n ) × R n × Tm,n → 2 R . Then, we have the following statements, which are closely related to the results presented in [15] for tensor eigenvalue complementarity problems.

Stability analysis
In this subsection, we study the sensitivity of TCPs at an isolated solution. Consider the TCP(K,q,Ā) with a given solutionx which is assumed to be locally unique, i.e., there exists a neighborhood V ofx such that SOL(K,q,Ā) ∩ V = {x}.
The cone K ∈ C(R n ) is fixed throughout the rest of this section. Like Section 5.1, we also assume no special structure on K other than the fact that it is closed and convex. We want to study the change ofx as (q,Ā) is perturbed. A central question is of course whether the perturbed TCP(K, q, A), where (q, A) is a small perturbation of (q,Ā), will have a solution that is nearx. To answer this question, we first present the following proposition.
Proposition 5.1 Let K ∈ C(R n ), A ∈ Tm,n and q ∈ R n . Ifx ∈ SOL(K, q, A) is isolated, then for every neighborhood N ofx satisfying SOL(K, q, A) ∩ cl(N ) = {x} (5.4) and for every sequence of {(A l , q (l) )} converging to (A, q), every sequence of vectors {x (l) }, where x (l) ∈ SOL(K, q (l) , A l ) ∩ N for every l, converges tox.
Proof Note that such a sequence {x (l) } in the proposition must be bounded. Let x * be any accumulation point of this sequence, which means that there exists )} converges to (A, q), by a simple limiting argument, it is easy to know that x * ∈ SOL(K, q, A) ∩ cl(N ). By (5.4), we know that x * =x. Therefore, {x (l) } converges tox as l → ∞.

⊓ ⊔
Recall that the tangent cone of K at a vectorx ∈ K, denoted by T (x, K), consists of all vectors v ∈ R n , for which there exists a sequence of vectors {x (k) } ⊂ K and a sequence of positive scalars {τ k } such that For givenx ∈ R n , denote Proposition 5.2 Let K ∈ C(R n ), A ∈ Tm,n and q ∈ R n . Suppose that A is subsymmetric with respect to the indices {i 2 , . . . , im}. Ifx ∈ SOL(K, q, A) and n i1,i2,...,im=1
Proof Assume for the sake of contradiction thatx is not locally unique. There exists a sequence {x (l) } ⊂ SOL(K, q, A) converging tox with x (l) =x for all l.
By the definition of T (x, K), it is easy to see that every accumulation point of the normalized sequence {(x (l) −x)/ x (l) −x } must belong to T (x, K). Let v be any such point. Without loss of generality, we assume ( Sincex and x (l) ∈ SOL(K, q, A), we have By (5.6) and (5.7), we havex and (x (l) ) ⊤ Ax m−1 + q ≥ 0 (5.9) for every l. Consequently, by (5.6) and (5.9), we have respect to the indices {i 2 , . . . , im}, by (5.12) and the fact that the function Ax m−1 is continuously differentiable atx, we know that for every l, where the last inequality is due to (5.10). Diving by x (l) −x in the above expression and letting l → ∞, we obtain n i1,i2,...,im=1 which is a contradiction. The proof is completed.

⊓ ⊔
We are now at the stage of extending Theorem 7.3.12 of [4] to tensors, which is the main stability result of the problem under consideration.
Consequently, sinceĀ is sub-symmetric with respect to the indices {i 2 , . . . , im}, it holds that for every l, where the second equality comes from (5.15) and the boundedness of {x (l) }, and the third equality is due to the fact thatĀx m−1 is continuously differentiable at x. By diving by x (l) −x 2 in the above expression and letting l → ∞, we know v ⊤ Ax m−2 v ≤ 0.
It is a contradiction. The proof is completed.

Conclusions
We consider the TCP over a general cone, which is an interesting generalization of LCP studied deeply in the literature. Unfortunately, the results of LCPs are not directly applicable to TCPs due to the nonlinearity of TCPs. However, as a special case of NCPs, in this paper, we first derive some specific forms of the solution existence results for TCP(K, q, A) from the results presented in [9]. In particular, when the general cone K reduces to a nonnegative cone R n + as discussed in the literature, we show the existence of a solution of TCP(R n + , q, A) under copositivity, which is a weaker condition than the strict copositivity in previous papers. Moreover, we investigate the topological properties of the solution set SOL(K, q, A) and stability of TCP(K, q, A) at a given solution. These results are new and further enrich the theory of TCPs.