1 Introduction

The classical linear complementarity problems (LCPs) have wide range of applications in applied science and technology [1], such as operations research, economics, engineering, to just name a few. The nonlinear complementarity problems (NCPs) are generalizations of LCPs by considering general nonlinear mappings [2]. Likewise, the so-dubbed quadratic complementarity problems (QCPs) that will be studied in this article are generalizations of LCPs by considering quadratic polynomial mappings on the one hand and more concrete realizations of NCPs on the other hand.

One motivation for studying QCPs is the three-person noncooperative game [3, 4]; another is a generalized Markowitz portfolio problem whose first-order optimality condition is a QCP [5]. The generalized Markowitz portfolio problem is in general NP-hard, and therefore properties on its KKT points are helpful and would be guidelines to design-efficient algorithms to solve it.

Though QCPs form a subclass of NCPs [2], they deserve particular investigations with at least twofold reasons: (i) they would serve as a bridge between the LCPs and the general NCPs, a very first step toward the nonlinear cases, and of which a concrete case; and (ii) they give a unified model for several classes of optimization problems (e.g., the cubic polynomial minimizations over the nonnegative orthant, the generalized Markowitz portfolio problems and three-person noncooperative games), which should have their own specifically developed theory and numerical methodologies. Actually, our research will show that the study on QCPs can even broaden the boundary of the knowledge for NCPs (cf. Sects. 3.2, 4).

This study also comes from the recent trend on tensor complementarity problems (TCPs). QCPs encompass the third-order TCPs in which the nonlinear mappings are the sum of quadratic forms and constant vectors [6]. In this field, Song and Qi showed the existence of solutions for TCPs under (strict) semi-positivity [6] and presented some relations among several classes of tensors [7]. Che et al. [8] studied properties of TCPs with positive-definite tensors. Song and Yu [9] further studied S tensors and properties of the solution sets of the corresponding TCPs. Bai et al. [10] showed that solution sets of TCPs with P tensors are nonempty and compact. Huang and Qi [3] reformulated a class of multilinear games as TCPs, providing examples for TCPs and establishing a bridge between these two classes of problems. For more research in this field and related, please refer to [6,7,8, 11, 12] and references therein.

This article will give a study on solution sets of QCPs using various tools from classical NCPs, tensor analysis, as well as some particularly designed techniques. We will organize the rest of this article as follows.

Basic notation and concepts will be presented in Sect. 2. A generalized Frank–Wolfe theorem for cubic polynomial optimization problems will be given in Sect. 3.1, which involves \(R_0\) tensors. With this, existence of solutions to QCPs is given under mild assumptions. The compactness of the solution sets will be discussed in Sect. 3.2. C-strict copositivity of a tensor and K-positive semidefinite plus of a matrix will be introduced there, based on which a compactness result will be given. The result generalizes, actually broadens, the well-known ones in the literature [2, Proposition 2.2.12], and it is proven under a regularity condition (i.e., (15)), which is formulated geometrically. This regularity condition combines information on all of the tensor, the matrix and the vector. Examples will be presented to show these promised novelties.

Uniqueness of the solution set will be investigated in Sect. 4 in terms of the null spaces of a collection of matrices. The uniqueness theorem involves the above regularity condition. The result generalizes and broadens the literature—it can handle a tensor which is not strictly copositive. An example is given there. Final remarks are given in the last section to conclude this article.

2 Preliminaries

In this article, \(\mathbb {R}^n\) denotes the Euclidean space of n-vectors of real numbers with standard inner product \(\langle \cdot ,\cdot \rangle \) and its induced Euclidean norm \(\Vert \cdot \Vert \). The nonnegative orthant of \(\mathbb {R}^n\) is denoted by \(\mathbb {R}^n_+\), whose interior is denoted by \({\mathbb {R}}^n_{++}\). \({\mathbb {R}}^n_-\) and \({\mathbb {R}}^n_{--}\) are defined similarly with respect to the nonpositive orthant. The symbol \(\mathbf 0\) denotes the zero vector in \(\mathbb {R}^n\). Lowercase bold letters (such as \(\mathbf x\), \(\mathbf y\)) represent vectors, capital letters (such as A, B) for matrices, and calligraphic letters (such as \(\mathcal A\), \(\mathcal B\)) for tensors.

A (real) third-order n-dimensional tensor (a.k.a. hypermatrix [13]), written as \({\mathcal {A}}=[a_{i_{1}i_2 i_3}]_{i_1,i_2,i_3=1}^n\in \mathbb {R}^{n\times n\times n}\), is a third-way array, where \(i_{j}\in \{1, \ldots , n\}\) and \(j\in \{1,2,3\}\). The set of all third-order n-dimensional tensors is denoted by \({\text {T}}({\mathbb {R}}^n,3)\), and the set of all \(n\times n\) (symmetric) matrices is denoted by (\({\text {S}}({\mathbb {R}}^n,2)\)) \({\text {T}}({\mathbb {R}}^n,2)\). The inner product on \({\text {T}}({\mathbb {R}}^n,3)\), denoted also as \(\langle \cdot ,\cdot \rangle \), is defined as follows: for any \({\mathcal {A}}=[a_{ijk}]_{i,j,k=1}^n,\mathcal B=[b_{ijk}]_{i,j,k=1}^n\in {\text {T}}({\mathbb {R}}^n,3)\)

$$\begin{aligned} \langle \mathcal A,\mathcal B\rangle :=\sum _{i=1}^n\sum _{j=1}^n\sum _{k=1}^na_{ijk}b_{ijk}. \end{aligned}$$

The inner product on \({\text {T}}({\mathbb {R}}^n,2)\) can be defined similarly. A tensor \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) can be viewed as a concatenation of n matrices of size \(n\times n\). More precisely, for all \(i\in \{1,\dots ,n\}\), the ith slice \({\mathcal {A}}_{i,\cdot ,\cdot }\) of \({\mathcal {A}}\) refers to the matrix \([a_{ijk}]_{j,k=1}^n\).

Given vectors \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\in {\mathbb {R}}^n\), \(\mathbf{x}\otimes \mathbf{y}\otimes \mathbf{z}\) refers to a rank-one tensor whose (i, j, k)th component is \(x_{i}y_{j}z_{k}\). \(\mathbf{x}^{\otimes 3}\) simplifies the symmetric rank-one tensor \(\mathbf{x}\otimes \mathbf{x}\otimes \mathbf{x}\). \(\mathbf{x}^{\otimes 2}\) is defined similarly.

Let \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^{n}, 3)\) and \(\mathbf{x}\in {\mathbb {R}}^n\), \({\mathcal {A}}\mathbf{x}^{2}\) is a vector with its ith component as

$$\begin{aligned} ({\mathcal {A}}\mathbf{x}^{2})_{i}:=\sum \limits _{i_{2},i_{3}=1}^{n}a_{ii_{2}i_{3}}x_{i_{2}}x_{i_{3}},\ \text {for all } i\in \{1,\dots ,n\}. \end{aligned}$$

A tensor \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^{n}, 3)\) is copositive, if \( {\mathcal {A}}\mathbf{x}^3:=\mathbf{x}^\mathsf {T}({\mathcal {A}}\mathbf{x}^2)\ge 0\ \text {for all }\mathbf{x}\in {\mathbb {R}}^n_+. \) It is called strictly copositive, if \( {\mathcal {A}}\mathbf{x}^3>0\ \text {for all }\mathbf{x}\in {\mathbb {R}}^n_+{\setminus }\{\mathbf 0\}. \)

The QCP refers to finding a vector \(\mathbf{x}\in {\mathbb {R}}^n\) such that

$$\begin{aligned}&\text {(QCP)}\ \ \ \ \mathbf{x}\ge \mathbf {0},\ {\mathcal {A}}\mathbf{x}^2+B\mathbf{x}+\mathbf c\ge \mathbf {0}\ \text {and }\mathbf{x}^\mathsf {T}({\mathcal {A}}\mathbf{x}^2+B\mathbf{x}+\mathbf c)=0,&\end{aligned}$$
(1)

in which \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) a given third-order tensor, \(B\in {\mathbb {R}}^{n\times n}\) a given matrix, and \(\mathbf c\in {\mathbb {R}}^n\) a given vector; \(\mathbf x\ge \mathbf 0\) means \(x_i\ge 0\) for all \(i\in \{1,\dots ,n\}\).

For QCP (1), sometimes it is without loss of any generality to consider tensors in the subspace \({\mathbb {R}}^n\otimes {\text {S}}({\mathbb {R}}^n,2)\) of \({\text {T}}({\mathbb {R}}^n,3)\). It is the set of tensors, which have symmetric elements with respect to the second and the third indices, i.e., \({\mathcal {A}}\in {\mathbb {R}}^n\otimes {\text {S}}({\mathbb {R}}^n,2)\) means \( {\mathcal {A}}_{i,\cdot ,\cdot }\in {\text {S}}({\mathbb {R}}^n,2)\) for all \(i\in \{1,\dots ,n\}\). Denote by \( A_i:={\mathcal {A}}_{i,\cdot ,\cdot }\ \text {for all }i\in \{1,\dots ,n\}. \) Then, with \( F(\mathbf{x}):={\mathcal {A}}\mathbf{x}^2+B\mathbf{x}+\mathbf c, \) we have

$$\begin{aligned} \nabla (\mathbf{x}^\mathsf {T}F(\mathbf{x}))=(\mathbf{x}^\mathsf {T}A_1\mathbf{x},\dots , \mathbf{x}^\mathsf {T}A_n\mathbf{x})^\mathsf {T}+2\sum _{i=1}^nx_iA_i\mathbf{x}+(B+B^\mathsf {T})\mathbf{x}+\mathbf c. \end{aligned}$$
(2)

Given a tensor \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\), we define its transpose \({\mathcal {A}}^\mathsf {T}\) as the tensor in \({\text {T}}({\mathbb {R}}^n,3)\) with entries

$$\begin{aligned} ({\mathcal {A}}^\mathsf {T})_{ijk}=a_{jik}\ \text {for all }i,j,k\in \{1,\dots ,n\}. \end{aligned}$$

Denote by \( D_i:=\big ({\mathcal {A}}^\mathsf {T}\big )_{i,\cdot ,\cdot }\in {\mathbb {R}}^{n\times n}\ \text {for all }i\in \{1,\dots ,n\}, \) and define

$$\begin{aligned} {\mathcal {A}}^\mathsf {T}\mathbf{x}:=\sum _{i=1}^nx_iD_i\ \text {for all }\mathbf{x}\in {\mathbb {R}}^n. \end{aligned}$$

3 Nonemptiness and Compactness

In the following, we will denote the solution set of QCP (1) by \(\textsc {sol}({\mathcal {A}},B,\mathbf c)\).

3.1 Nonemptiness via a Frank–Wolfe-Type Theorem

A tensor \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) is called an \(R_0\) tensor [7], if the system

$$\begin{aligned} \mathbf{x}^\mathsf {T}A_i\mathbf{x}\ge 0,\ \text {if }x_i=0,\ \text {and } \mathbf{x}^\mathsf {T}A_i\mathbf{x}=0,\ \text {if }x_i>0\ \text {for all }i\in \{1,\dots ,n\} \end{aligned}$$

does not have a solution in \({\mathbb {R}}^n_+{\setminus }\{\mathbf{0}\}\).

Proposition 3.1

Let \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) be an \(R_0\) tensor. Then, whenever QCP (1) is feasible, the minimization problem

$$\begin{aligned} \inf \mathbf{x}^\mathsf {T}F(\mathbf{x})\ \text {s.t. }\mathbf{x}\ge \mathbf{0},\ F(\mathbf{x})\ge \mathbf{0} \end{aligned}$$
(3)

has an optimal solution.

Proof

Let the feasible solution set be \( S:=\{\mathbf{x}: \mathbf{x}\ge \mathbf{0},\ F(\mathbf{x})\ge \mathbf{0}\}. \) By assumption, \(S\ne \emptyset \). Denote by \( v_*=\inf \{\mathbf{x}^\mathsf {T}F(\mathbf{x}) : \mathbf{x}\in S\}. \) Let

$$\begin{aligned} B_\rho :=\{\mathbf{x}\in {\mathbb {R}}^n : \Vert \mathbf{x}\Vert \le \rho \} \end{aligned}$$

be the ball centered at zero with radius \(\rho \). Thus, there exists \(\kappa >0\) such that \( S\cap B_\rho \ne \emptyset \ \text {for all }\rho \ge \kappa . \) Let

$$\begin{aligned} v_\rho :=\min \{\mathbf{x}^\mathsf {T}F(\mathbf{x}) : \mathbf{x}\in S\cap B_\rho \}\ \text {for all }\rho \ge \kappa . \end{aligned}$$
(4)

The optimal solution set \(s_\rho \) of (4) is obvious compact. Thus, we define

$$\begin{aligned} \mathbf{x}_\rho \in {\text {argmin}}\{\Vert \mathbf{x}\Vert : \mathbf{x}\in s_\rho \}\ \text {for all }\rho \ge \kappa \end{aligned}$$

as a minimum norm optimal solution. We claim that there exists \(\gamma >\kappa \) such that

$$\begin{aligned} \Vert \mathbf{x}_\rho \Vert <\rho \ \text {for all }\rho \ge \gamma . \end{aligned}$$
(5)

Suppose, on the contrary, that there exists a sequence \(\{\rho _k\}\) such that

$$\begin{aligned} \Vert \mathbf{x}_{\rho _k}\Vert =\rho _k\ \text {with }\rho _k\rightarrow \infty . \end{aligned}$$

Taking subsequence if necessary, we may assume without loss of generality that \(\frac{\mathbf{x}_{\rho _k}}{\rho _k}\rightarrow \overline{\mathbf{x}}.\) Obviously, the feasibility and the normalization imply that

$$\begin{aligned} \overline{\mathbf{x}}\ge \mathbf{0}\ \text {and }\overline{\mathbf{x}}\ne \mathbf{0}. \end{aligned}$$
(6)

Dividing each defining polynomial of F by \(\rho _k^2\), we get with the feasibility that

$$\begin{aligned} \overline{\mathbf{x}}^\mathsf {T}A_i\overline{\mathbf{x}}\ge 0\ \text {for all }i\in \{1,\dots ,n\}. \end{aligned}$$
(7)

By the nonemptiness of S, we conclude that \( v_\rho \downarrow v_*. \) Thus,

$$\begin{aligned} v_{\rho _k}=\mathbf{x}_{\rho _k}^\mathsf {T}F(\mathbf{x}_{\rho _k})=\sum _{i=1}^n(\mathbf{x}_{\rho _k})_i \mathbf{x}_{\rho _k}^\mathsf {T}A_i\mathbf{x}_{\rho _k}+\mathbf{x}_{\rho _k}^\mathsf {T}B\mathbf{x}_{\rho _k}+\mathbf c^\mathsf {T}\mathbf{x}_{\rho _k} \end{aligned}$$

should be bounded. Dividing the equation by \(\rho _k^3\), we get that \({\mathcal {A}}\overline{\mathbf{x}}^3=0.\) This, together with (6) and (7), violates the hypothesis that \({\mathcal {A}}\) is an \(R_0\) tensor. Therefore, the claim (5) is proved.

In the next, we claim that there exists \(\tau >\gamma \) such that

$$\begin{aligned} v_{\rho }=v_*\ \text {for all }\rho \ge \tau . \end{aligned}$$
(8)

Suppose, on the contrary, that \(v_{\rho }>v_*\ \text {for all }\rho \ge \kappa .\) As \(v_{\rho }\downarrow v_*\), we can find \(\gamma<\rho _1<\rho _2\) such that \(v_{\rho _1}>v_{\rho _2}\). By the claim (5), we have that \( \Vert \mathbf{x}_{\rho _2}\Vert <\rho _2. \) Since \(v_{\rho _1}>v_{\rho _2}\), we have \( \Vert \mathbf{x}_{\rho _2}\Vert >\rho _1. \) Let \(\rho _3=\Vert \mathbf{x}_{\rho _2}\Vert \). Then, \(\gamma<\rho _1<\rho _3<\rho _2\) and \(\Vert \mathbf{x}_{\rho _3}\Vert <\rho _3=\Vert \mathbf{x}_{\rho _2}\Vert .\) \(\rho _3<\rho _2\) implies that \(v_{\rho _2}\le v_{\rho _3}\).

If \(v_{\rho _2}=v_{\rho _3}\), then \(\mathbf{x}_{\rho _3}\) is an optimal solution to \(v_{\rho _2}\) with a strictly smaller norm than \(\mathbf{x}_{\rho _2}\), which is a contradiction to the definition. If \(v_{\rho _2}<v_{\rho _3}\), then it, together with \(\rho _3=\Vert \mathbf{x}_{\rho _2}\Vert \), gives a contradiction to \(\mathbf{x}_{\rho _3}\) being an optimal solution. As a consequence, the claim (8) is proved. Thus, the proposition follows immediately. \(\square \)

Proposition 3.1 is a generalization of the classical Frank–Wolfe theorem [14].

Definition 3.1

(Co-semidefinite pair) A tensor \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) and a matrix \(B\in {\text {T}}({\mathbb {R}}^n,2)\) is called a co-semidefinite pair, if the matrix pencil \({\mathcal {A}}^\mathsf {T}\mathbf{x}+B\) is positive semidefinite for all \(\mathbf{x}\in {\mathbb {R}}^n_+\).

Obviously, \({\mathcal {A}}\) and B form a co-semidefinite pair if and only if B, \(D_1\), \(\dots \), \(D_n\) are all positive semidefinite matrices. Actually, the sufficiency is immediate. For the necessity, the matrix B should be positive semidefinite is apparent. For the positive semidefiniteness of the matrices \(D_i\)s, we can drive a proof by contradiction. Suppose \(D_1\) is not positive semidefinite, then with sufficiently large t, the matrix \({\mathcal {A}}^\mathsf {T}(t\mathbf e_1)+B\) would not be positive semidefinite.

Proposition 3.2

Let \(\mathbf{x}_*\) be an optimal solution of (3) with \({\mathcal {A}}\in {\mathbb {R}}^n\otimes {\text {S}}({\mathbb {R}}^n,2)\). If a constraint qualification is satisfied at \(\mathbf{x}_*\) to ensure at which the Karush–Kuhn–Tucker condition holds, and \({\mathcal {A}}\) and B form a co-semidefinite pair, then \(\mathbf{x}_*\in \textsc {sol}({\mathcal {A}},B,\mathbf c)\).

Proof

It follows from the Karush-Kuhn-Tucker condition hypothesis that there exists \(\mathbf{u}_*\ge \mathbf{0}\) such that (cf. (2) for gradients)

$$\begin{aligned}&\begin{bmatrix} \mathbf{x}_*^\mathsf {T}A_1\mathbf{x}_*\\ \vdots \\ \mathbf{x}_*^\mathsf {T}A_n\mathbf{x}_*\end{bmatrix}+2\sum _{i=1}^n(\mathbf{x}_*)_iA_i \mathbf{x}_*+(B+B^\mathsf {T})\mathbf{x}_*+\mathbf c-2\sum _{i=1}^n(\mathbf u_*)_iA_i\mathbf{x}_*-B^\mathsf {T}\mathbf{u}_*\ge \mathbf{0},\nonumber \\ \end{aligned}$$
(9)
$$\begin{aligned}&\quad \mathbf{x}_*^\mathsf {T}\left( \begin{bmatrix}\mathbf{x}_*^\mathsf {T}A_1\mathbf{x}_*\\ \vdots \\ \mathbf{x}_*^\mathsf {T}A_n\mathbf{x}_*\end{bmatrix}+ 2\sum _{i=1}^n(\mathbf{x}_*)_iA_i\mathbf{x}_* +(B+B^\mathsf {T})\mathbf{x}_*\right. \nonumber \\&\qquad \left. +\,\mathbf c-2\sum _{i=1}^n(\mathbf u_*)_iA_i\mathbf{x}_*-B^\mathsf {T}\mathbf{u}_*\right) =0, \end{aligned}$$
(10)
$$\begin{aligned}&\quad \mathbf{u}_*^\mathsf {T}F(\mathbf{x}_*)=0,\ \mathbf{x}_*\ge \mathbf{0},\ \mathbf{u}_*\ge \mathbf{0}.\ F(\mathbf{x}_*)\ge \mathbf{0}. \end{aligned}$$
(11)

Thus, multiplying (9) by \(\mathbf{u}_*\) and using the complementarity of \(\mathbf{u}_*\) (cf. (11)), we have

$$\begin{aligned} \mathbf{u}_*^\mathsf {T}\left( -2\sum _{i=1}^n(x_*)_iA_i\mathbf{x}_*-B^\mathsf {T}\mathbf{x}_* +2\sum _{i=1}^n(u_*)_iA_i\mathbf{x}_*+B^\mathsf {T}\mathbf{u}_*\right) \le 0, \end{aligned}$$

and, using the feasibility of \(\mathbf{x}_*\), we have from (10) that

$$\begin{aligned} \mathbf{x}_*^\mathsf {T}\left( 2\sum _{i=1}^n(x_*)_iA_i\mathbf{x}_*+B^\mathsf {T}\mathbf{x}_*-2 \sum _{i=1}^n(u_*)_iA_i\mathbf{x}_*-B^\mathsf {T}\mathbf{u}_*\right) \le 0. \end{aligned}$$
(12)

Therefore,

$$\begin{aligned} 2\langle {\mathcal {A}}, \mathbf{u}_*\otimes \mathbf{u}_*\otimes \mathbf{x}_*-\mathbf{x}_*\otimes \mathbf{u}_*\otimes \mathbf{x}_*+\mathbf{x}_*^{\otimes 3}-\mathbf{u}_*\otimes \mathbf{x}_*\otimes \mathbf{x}_*\rangle +\langle B,(\mathbf{u}_*-\mathbf{x}_*)^{\otimes 2}\rangle \le 0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} 2\langle {\mathcal {A}}^\mathsf {T}\mathbf{x}_*,(\mathbf{x}_*-\mathbf{u}_*)^{\otimes 2}\rangle +\langle B,(\mathbf{u}_*-\mathbf{x}_*)^{\otimes 2}\rangle \le 0. \end{aligned}$$

Since \({\mathcal {A}}\) and B form a co-semidefinite pair, we have

$$\begin{aligned} 2\langle {\mathcal {A}}^\mathsf {T}\mathbf{x}_*,(\mathbf{x}_*-\mathbf{u}_*)^{\otimes 2}\rangle +\langle B,(\mathbf{u}_*-\mathbf{x}_*)^{\otimes 2}\rangle =0. \end{aligned}$$

Thus, (12) should be an equality, which together with (10), further implies that \(\mathbf{x}_*^\mathsf {T}F(\mathbf{x}_*)=0\). Thus, \(\mathbf{x}_*\in \textsc {sol}({\mathcal {A}},B,\mathbf c)\). \(\square \)

Whenever the optimal value of (3) is zero, we have complementarity of \(\mathbf{x}_*\) and \(F(\mathbf{x}_*)\). Recall that \(\nabla F(\mathbf{x})=2[A_1\mathbf{x},\dots ,A_n\mathbf{x}]^\mathsf {T}+B\). Therefore, the linear independence constraint qualification (LICQ) holds generically. Actually, the LICQ holds generically, if we can find a concrete example such that the matrix \(\nabla F(\mathbf{x})\) is of full rank by Hilbert’s Nullstellensatz. This is easy to see, with \(\mathcal A=0\) being the zero tensor, B an arbitrary matrix of full rank, and \(\mathbf c\ge 0\). Then, \(\mathbf 0\) is an optimizer to problem (3), while \(\nabla F(\mathbf 0)=B\), which is of full rank.

If \({\mathcal {A}}\in {\mathbb {R}}^n\otimes {\text {S}}({\mathbb {R}}^n,2)\) is an \(R_0\) tensor, then Proposition 3.1 guarantees an optimal solution for (3) whenever (1) is feasible. The following example comes from Propositions 3.1 and  3.2.

Example 3.1

Let tensor \({\mathcal {A}}\in {\mathbb {R}}^2\otimes {\text {S}}({\mathbb {R}}^2,2)\) with \(a_{111}=a_{222}=1\) and all other entries \(a_{i_{1}i_{2}i_{3}}=0\), and matrix \(B=\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\). It is easy to verify that \(\mathcal A\) is an \(R_{0}\) tensor, \(D_{1}=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}\), \(D_{2}=\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\) and \(B=\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\) are all positive semidefinite matrices, and hence \(\mathcal A\) and B form a co-semidefinite pair. It is also easy to see that the corresponding QCP (1) is feasible for any \(\mathbf c\in {\mathbb {R}}^2\). The LICQ holds at any optimal solution in this case. By Proposition 3.1, the solution set is nonempty.

For this example, the nonemptiness can also be checked by direct calculation. Actually, if \(c_1\ge 0\), then we can take \(x_1=0\); otherwise, \(x_1=\sqrt{-c_1}>0\). If \(c_2\ge 0\), then we can take \(x_2=0\) as well; otherwise, \(x_2=\frac{\sqrt{1-4c_2}-1}{2}>0\).

3.2 Compact Solution Sets

In this section, we will study the compactness/existence of the solution set \(\textsc {sol}({\mathcal {A}},B,\mathbf c)\). Obviously, the set \(\textsc {sol}({\mathcal {A}},B,\mathbf c)\) is closed by continuity.

Definition 3.2

(C-strict copositivity) Let \(C\subseteq {\mathbb {R}}^n_+\) be a nonempty closed cone. A tensor \(\mathcal A\in {\text {T}}({\mathbb {R}}^n,3) \) is called C -strictly copositive, if \(\mathcal A\) is copositive and strictly copositive on the cone C, i.e.,

$$\begin{aligned} \mathcal A\mathbf x^3\ge 0\ \text {for all }\mathbf x\in {\mathbb {R}}^n_+,\ \text {and }\mathcal A\mathbf x^3>0\ \text {for all }\mathbf x\in C{\setminus }\{\mathbf 0\}. \end{aligned}$$

Obviously, \(\{\mathbf 0\}\)-strict copositivity is the copositivity and \({\mathbb {R}}^n_+\)-strict copositivity is the strict copositivity in the usual sense, respectively. There are plenty of examples of C-strictly copositive tensors with C being a face of \({\mathbb {R}}^n_+\), i.e.,

$$\begin{aligned} C=\{\mathbf x\in {\mathbb {R}}^n_+ : x_i=0\ \text {for all }i\in I\} \end{aligned}$$

for some subset \(I\subseteq \{1,\dots ,n\}\). In the following, we give an example with C being not a face of \({\mathbb {R}}^n_+\).

Example 3.2

Let \(\mathcal {A}=(a_{i_{1}i_{2}i_{3}})\in {\text {T}}({\mathbb {R}}^2,3)\) and \(a_{111}=1\), \(a_{112}=-1\), \(a_{211}=1\) and \(a_{i_{1}i_{2}i_{3}}=0\) for all the other \(i_1,i_2,i_3\in \{1,2\}\). Let

$$\begin{aligned} C:=\mathbb {R}_{\ge }^{2}=\{\mathbf{x}\in \mathbb {R}_{+}^{2} : x_{1}\ge x_{2}\} \end{aligned}$$

be the cone of vectors with nonincreasing components. Then, \(\mathcal {A}\) is C-strictly copositive. Actually, \({\mathcal {A}}\mathbf{x}^3=x_1^3\). Clearly, \(\mathcal {A}\) is copositive and strictly copositive on the cone C, while \({\mathcal {A}}\) is not strictly copositive.

We remark that checking C-strict copositivity shall be an NP-hard problem in the general setting, since it includes the usual strict copositivity problem. While, when C is a polyhedral or more general a semi-algebraic set defined by explicit polynomial equalities/inequalities, standard SDP hierarchy relaxation techniques in polynomial optimization can be applied. This shall be studied in the next research to keep the main scope of this article on solution sets of QCPs.

Definition 3.3

(K-positive semidefinite plus) Let \(K\subseteq {\mathbb {R}}^n\) be a nonempty closed cone. A matrix \(B\in {\mathbb {R}}^{n\times n}\) is called K -positive semidefinite plus, if B is positive semidefinite plus on K, i.e.,

$$\begin{aligned} \mathbf x^\mathsf {T}B\mathbf x\ge & {} 0,\ \text {for all }\mathbf x\in K, \ \text {and }\nonumber \\ \text {whenever } \mathbf x^\mathsf {T}B\mathbf x= & {} 0\ \text {for }\mathbf x\in K,\ \ \text {it follows } B\mathbf x=\mathbf 0. \end{aligned}$$
(13)

In Definition 3.3, the subset K can be a linear subspace. If \(K={\mathbb {R}}^n\), we call B simply positive semidefinite plus. Given a point \(\mathbf x\in {\mathbb {R}}^n\), \({\mathbb {R}}_+\mathbf x\) is the cone \(\{\alpha \mathbf x : \alpha \in {\mathbb {R}}_+\}\), and

$$\begin{aligned} ({\mathbb {R}}_+\mathbf x)^\diamond :=\left\{ \begin{array}{ll}({\mathbb {R}}_+\mathbf x)^*,&{}\quad \text {if }\mathbf x\ne \mathbf 0,\\ \{\mathbf 0\},&{}\quad \text {otherwise},\end{array}\right. \end{aligned}$$
(14)

where \(K^*\) means the dual cone of a given cone K. \(K^\complement :={\mathbb {R}}^n{\setminus } K\) is the complement of K in \({\mathbb {R}}^n\).

Proposition 3.3

(Compact solution set) Let \(\mathcal {A}\in {\text {T}}({\mathbb {R}}^n,3) \) be C-strictly copositive for a nonempty closed cone \(C\subseteq {\mathbb {R}}^n_+\), \(B\in \mathbb {R}^{n\times n}\) a K-positive semidefinite plus matrix for a nonempty closed cone \(K\subseteq {\mathbb {R}}^n\), and \(\mathbf c\in \mathbb {R}^{n}\) a vector. Let the intersection of the kernel of B and the linear subspace \({\text {lin}}(K)\) generated by K be \(L\subseteq {\mathbb {R}}^n\). Suppose that \(K^\complement \cap {\mathbb {R}}^n_+\subseteq C\), and

$$\begin{aligned} L\cap {\mathbb {R}}^n_+\subseteq C\cup \bigg [{\text {int}}\big (({\mathbb {R}}_+\mathbf c)^\diamond \big )\cap {\mathbb {R}}^n_+\bigg ]. \end{aligned}$$
(15)

Then, the QCP (1) has a nonempty and bounded solution set.

Proof

It is sufficient to show that the set

$$\begin{aligned} L_{\le }:=\{\mathbf{x}\in {\mathbb {R}}^n_+ : \mathbf{x}^\mathsf {T}(\mathbf c+B\mathbf{x}+\mathcal {A}\mathbf{x}^{2})\le 0\} \end{aligned}$$

is bounded, by [2, Proposition 2.2.3]. In the following, we assume, on the contrary, that the set \(L_{\le }\) is unbounded and from which we will derive a contradiction.

Suppose that \(\{\mathbf{x}^{k}\}\subseteq L_{\le }\) is an unbounded sequence. Without loss of generality, we assume that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \mathbf{x}^{k}\Vert =\infty ,~~\lim _{k\rightarrow \infty } \frac{\mathbf{x}^{k}}{\Vert \mathbf{x}^{k}\Vert }=\mathbf d\in {\mathbb {R}}^n_+. \end{aligned}$$

Obviously, \( \mathbf 0\ne \mathbf d\ge \mathbf 0 \). Further taking subsequence if necessary, we can assume that either

$$\begin{aligned} \{\mathbf x^k\}\subset K,\ \text {or } \{\mathbf x^k\}\subset K^\complement . \end{aligned}$$

Since C is closed, it follows from \(K^\complement \cap {\mathbb {R}}^n_+\subseteq C\) that if \(\{\mathbf x^k\}\subset K^\complement \), then \(\mathbf d\in C\).

On the other hand, it follows from

$$\begin{aligned} (\mathbf{x}^{k})^\mathsf {T}(\mathbf c+B\mathbf{x}^{k}+\mathcal {A}(\mathbf{x}^{k})^{2})\le 0, \end{aligned}$$
(16)

and the copositivity of \({\mathcal {A}}\) that \(\mathcal {A}\mathbf d^{3}=0.\) Thus,

$$\begin{aligned} \mathbf d\notin C \end{aligned}$$
(17)

by the C-strict copositivity of \({\mathcal {A}}\). Consequently,

$$\begin{aligned} \{\mathbf x^k\}\subset K, \end{aligned}$$
(18)

and henceforth

$$\begin{aligned} \mathbf d\in K\cap {\mathbb {R}}^n_+. \end{aligned}$$
(19)

It follows from (16) and the copositivity of \({\mathcal {A}}\) that

$$\begin{aligned} (\mathbf{x}^{k})^\mathsf {T}(\mathbf c+B\mathbf{x}^{k})\le 0,\ \text {for all }k. \end{aligned}$$
(20)

Therefore, by dividing the inequality by \(\Vert \mathbf{x}^k\Vert ^2\) and taking limitation, we conclude that \(\mathbf d^\mathsf {T}B\mathbf d\le 0,\) which, together with (19) and the fact that B is K-positive semidefinite plus, implies that

$$\begin{aligned} B\mathbf d=0,\ \text {or equivalently }\mathbf d\in L. \end{aligned}$$

Thus,

$$\begin{aligned} \mathbf d\in L\cap {\mathbb {R}}^n_+. \end{aligned}$$
(21)

By (18) and (20), as well as the K-positive semidefiniteness plus of B, we have that \(\mathbf c^\mathsf {T}\mathbf d\le 0,\) which implies that \( \mathbf d\notin {\text {int}}\big (({\mathbb {R}}_+\mathbf c)^\diamond \big ). \)

This, together with (17) and (21), gives a contradiction to (15). \(\square \)

Condition (15) is called a regularity for QCP (1).

For a linear subspace \(L\subseteq {\mathbb {R}}^n\), \(L^\mathsf {\perp }\) is the orthogonally complementary subspace of L in \({\mathbb {R}}^n\).

Proposition 3.4

Let \(B\in \mathbb {R}^{n\times n}\) be a positive semidefinite plus matrix with the kernel being \(L\subseteq {\mathbb {R}}^n\), and \(\mathbf c\in \mathbb {R}^{n}\) be a vector. Then, the following statements are equivalent:

(a) :

There exists a \(\mathbf y\in {\mathbb {R}}^n\) such that

$$\begin{aligned} \mathbf c+B\mathbf y\in {\mathbb {R}}^n_{++}. \end{aligned}$$
(22)
(b) :

The following regularity holds

$$\begin{aligned} L\cap {\mathbb {R}}^n_+\subseteq \{\mathbf 0\}\cup \big [{\text {int}}\big (({\mathbb {R}}_+\mathbf c)^\diamond \big )\cap {\mathbb {R}}^n_+\big ]. \end{aligned}$$
(23)

Proof

Suppose that there exists a \(\mathbf y\in {\mathbb {R}}^n\) such that \(\mathbf c+B\mathbf y\in {\mathbb {R}}^n_{++}\). Then,

$$\begin{aligned} \mathbf c\in L^\perp +{\mathbb {R}}^n_{++}, \end{aligned}$$

since the range space of B is \(L^\perp \). Thus, from [15] we have

$$\begin{aligned} \mathbf c\in {\text {int}}((L\cap {\mathbb {R}}^n_+)^*)={\text {int}}(L^\perp +{\mathbb {R}}^n_{++})=L^\perp +{\mathbb {R}}^n_{++}, \end{aligned}$$

which implies

$$\begin{aligned} \langle \mathbf c,\mathbf d\rangle >0\ \text {for all }\mathbf d\in L\cap {\mathbb {R}}^n_+{\setminus }\{\mathbf 0\}. \end{aligned}$$

Therefore, either \(L\cap {\mathbb {R}}^n_+=\{\mathbf 0\}\) or

$$\begin{aligned} L\cap {\mathbb {R}}^n_+{\setminus }\{\mathbf 0\}\subseteq \big [{\text {int}}\big (({\mathbb {R}}_+\mathbf c)^\diamond \big )\cap {\mathbb {R}}^n_+\big ], \end{aligned}$$

while both cases are covered by (23).

If \(L\cap {\mathbb {R}}^n_+=\{\mathbf 0\}\), then we have from [15] that

$$\begin{aligned} {\mathbb {R}}^n=(L\cap {\mathbb {R}}^n_+)^*=L^\perp +{\mathbb {R}}^n_+. \end{aligned}$$

We must have a point \(\mathbf z\in {\mathbb {R}}^n\) such that \( B\mathbf{z}\in {\mathbb {R}}^n_{--}. \) For any \(\mathbf c\in {\mathbb {R}}^n\), we can find a \(\mathbf y\in {\mathbb {R}}^n\) such that \(\mathbf c+B\mathbf y\in {\mathbb {R}}^n_{+}\). Therefore,

$$\begin{aligned} \mathbf c+B(\mathbf y-\mathbf{z})\in {\mathbb {R}}^n_{++}. \end{aligned}$$

So, (22) follows.

In the following, suppose that \(L\cap {\mathbb {R}}^n_+\) has dimension being strictly larger than zero. The condition (23) is equivalent to

$$\begin{aligned} \langle \mathbf c,\mathbf d\rangle >0\ \text {for all }\mathbf d\in L\cap {\mathbb {R}}^n_+{\setminus }\{\mathbf 0\}. \end{aligned}$$

Therefore,

$$\begin{aligned} \mathbf c\in {\text {int}}((L\cap {\mathbb {R}}^n_+)^*)={\text {int}}(L^\perp +{\mathbb {R}}^n_{++})=L^\perp +{\mathbb {R}}^n_{++}. \end{aligned}$$

Consequently, we have that there exists a \(\mathbf y\in {\mathbb {R}}^n\) such that \(\mathbf c+B\mathbf y\in {\mathbb {R}}^n_{++}\). \(\square \)

We know from Proposition 3.4 that (22) is equivalent to a special case (23) of (15). Thus, Proposition 3.3 generalizes [2, Proposition 2.2.12] and [8, Theorem 4.5(b)] for TCPs with third-order tensors:

  1. (i)

    Taking \(C=\{\mathbf 0\}\) in Proposition 3.3, together with Proposition  3.4, Proposition 3.3 generalizes [2, Proposition 2.2.12(a)].

  2. (ii)

    Taking \(C=\mathbb {R}^{n}_{+}\) in Proposition 3.3, \(\mathcal {A}\) is then strictly copositive, which implies that the coercivity condition in [2, Proposition 2.2.12(b)] is satisfied, then Proposition 3.3 generalizes [2, Proposition 2.2.12(b)].

  3. (iii)

    Taking \(C={\mathbb {R}}^n_{+}\) and \(B=0\) in Proposition 3.3, we recover [8, Theorem 4.5(b)] for TCPs with third-order tensors, since (15) is satisfied for any given \(\mathbf c\).

Whenever C is a nontrivial proper subcone in \({\mathbb {R}}^n_+\), the standard NCP and TCP theory is helpless, while Proposition 3.3 may provide a solution.

Example 3.3

Let \(\mathcal {A}=(a_{i_{1}i_{2}i_{3}})\in {\text {T}}({\mathbb {R}}^2,3)\) and \(a_{122}=-1\), \(a_{221}=1\), \(a_{222}=1\) and \(a_{i_{1}i_{2}i_{3}}=0\) for all the other \(i_1,i_2,i_3\in \{1,2\}\), \(B=\begin{bmatrix}2&0\\ 0&0\end{bmatrix}\). Let

$$\begin{aligned} C=\{\mathbf{x}\in \mathbb {R}_{+}^{n} : x_{1}\le x_{2}\} \end{aligned}$$

and \(K=\mathbb {R}_{+}^{2}\). It can be checked that \(\mathcal A\) is C-strictly copositive, but fails to be strictly copositive; and B is K-positive semidefinite plus. With the notation as in Proposition 3.3, \(L=\{\mathbf x\in {\mathbb {R}}^2 : x_1=0\}\), and \(L\cap {\mathbb {R}}^2_+\subset C\). Thus, for any \(\mathbf c\in {\mathbb {R}}^2\), \(\textsc {sol}(\mathcal A, B, \mathbf c)\) is nonempty and compact.

We shall show the nonemptiness and compactness through direct calculation. If \(c_2\ge 0\), then \(x_2=0\), and hence \(2x_1^2+c_1x_1=0\). Thus, the solution set is nonempty (\(x_1=0\) is a solution) and always compact. If \(c_2<0\), there is a solution with \(x_1=0\) and \(x_2>0\) for any \(c_1\). On the other hand, in this case, solutions must be with \(x_2>0\). So, \(x_{2}^{2}+x_{1}x_{2}+c_{2}=0\). Since \(x_1\ge 0\), \(x_2\) cannot go to infinity. Thus, \(x_2\) is always bounded. If \(x_1\) goes to infinity, then \(x_2\) should go to zero to maintain the equality \(x_{2}^{2}+x_{1}x_{2}+c_{2}=0\), while \(-x_{2}^{2}+2x_{1}+c_{1}=0\) as \(x_1>0\), which is a contradiction for any fixed \(c_1\).

Next example is a modification of Example 3.3 in which \(\mathbf c\) plays a role.

Example 3.4

Let \(\mathcal {A}=(a_{i_{1}i_{2}i_{3}})\in {\text {T}}({\mathbb {R}}^2,3)\) and \(a_{122}=-1\), \(a_{221}=1\), and \(a_{i_{1}i_{2}i_{3}}=0\) for all the other \(i_1,i_2,i_3\in \{1,2\}\), the matrix B is as the previous one, and \(\mathbf c=(c_{1}, c_{2})^\mathsf {T}\) with some \(c_2>0\). Let \(C=\{\mathbf 0\}\), \(K=\mathbb {R}^2_+\).

In this case, \(L\cap \mathbb {R}^{2}_{+}=\{\mathbf{x}\in {\mathbb {R}}^2 : x_{1}=0, x_{2}\ge 0\}\) as well, while it is easy to see pictorially that \(L\cap \mathbb {R}^{n}_{+}\subset {\text {int}}\big (({\mathbb {R}}_+\mathbf c)^\diamond \big )\cap {\mathbb {R}}^2_+\). Thus, the regularity (15) holds as well and hence the solution set is nonempty and compact by Proposition 3.3. It can also be checked by direct calculation that the corresponding solution set is nonempty and compact in this case as Example 3.3.

Besides the above two illustrative examples for Proposition 3.3, we remark an application to the generalized Markowitz portfolio problem [5], whose details will be carried out in a next research. A forecast is that the tensor \({\mathcal {A}}\) is either copositive or C-strictly copositive for a cone of the form

$$\begin{aligned} {\mathbb {R}}^{n-4}_+\times \{0\}\times \{0\}\times \{0\}\times \{0\}\subset {\mathbb {R}}^{n}_+. \end{aligned}$$

In the following, we note a connection between the regularity (15) and the existence for solutions of QCP (1) via a generalized Frank–Wolfe theorem by Andronov et al. [16]. A tensor is symmetric, if the entries are invariant when permuting their indices.

The existence for solutions of QCP (1) with symmetric \({\mathcal {A}}\) and B satisfying (15) follows from this generalized Frank–Wolfe theorem for cubic polynomial objective under linear constraints. In this case, we consider the optimization problem

$$\begin{aligned} \min \bigg \{\frac{1}{3}{\mathcal {A}}\mathbf{x}^3+\frac{1}{2}\mathbf{x}^\mathsf {T}B\mathbf{x}+\mathbf c^\mathsf {T}\mathbf{x}: \mathbf{x}\ge \mathbf{0}\bigg \}. \end{aligned}$$
(24)

It is proved in [16] that whenever the objective function is bounded below over the feasible set, there is an optimal solution. Obviously, the LICQ holds at any optimal solution, which further implies the existence of a KKT point. It is straightforward to write down the KKT system for (24) and it is the same as QCP (1). Whenever the regularity (15) is satisfied, an almost the same proof as that for Proposition 3.3 will show that the objective function is indeed bounded from below.

4 Uniqueness

If \(A\in {\mathbb {R}}^{n\times n}\) is positive semidefinite, we define the null \({\text {null}}(A)\) of A as the set of vectors in \({\mathbb {R}}^n\) such that \(\mathbf{x}^\mathsf {T}A\mathbf{x}=0\), i.e.,

$$\begin{aligned} {\text {null}}(A):=\{\mathbf{x}\in {\mathbb {R}}^n : \mathbf{x}^\mathsf {T}A\mathbf{x}=0\}. \end{aligned}$$

It is easy to see that \({\text {null}}(A)\) is a linear subspace. If furthermore \(A\in {\text {S}}({\mathbb {R}}^n,2)\), i.e., A is symmetric, then \({\text {null}}(A)\) is the kernel of A, i.e.,

$$\begin{aligned} {\text {null}}(A)=\{\mathbf{x}\in {\mathbb {R}}^n : A\mathbf{x}=\mathbf 0\}, \end{aligned}$$

while in general this is not true.

Let \(\{D_1,\dots ,D_n\}\subset {\mathbb {R}}^{n\times n}\) be a collection of n matrices. If each \(D_i\) is positive semidefinite, then the nulls of the matrix pencils

$$\begin{aligned} x_1D_1+\dots +x_nD_n\ \text {for }\mathbf{x}\in {\mathbb {R}}^n_+ \end{aligned}$$

form a partially ordered finite set \({\text {W}}(D_1,\dots ,D_n)\). Actually, whenever each \(D_i\) is positive semidefinite, the null of the matrix \(x_1D_1+\dots +x_nD_n\) for \(\mathbf{x}\in {\mathbb {R}}^n_+\) is equal to the null of the matrix

$$\begin{aligned} {\text {sign}}(x_1)D_1+\dots +{\text {sign}}(x_n)D_n, \end{aligned}$$

where

$$\begin{aligned} {\text {sign}}(\alpha )=1 \ \ \text {if}\ \alpha >0; \ \text {or } \ {\text {sign}}(\alpha )=0 \ \ \text {if} \ \alpha =0; \ \text {or } \ {\text {sign}}(\alpha )=-1 \ \ \text {if}\ \alpha <0. \end{aligned}$$

In fact, the null of \({\text {sign}}(x_1)D_1+\dots +{\text {sign}}(x_n)D_n\) is equal to

$$\begin{aligned} \bigcap _{i=1}^n{\text {null}}\big ({\text {sign}}(x_i)D_i\big ). \end{aligned}$$

Therefore, with respect to the set inclusion, \({\text {W}}(D_1,\dots ,D_n)\) is a set with the maximal elements being \( \{{\text {null}}(D_i) : i\in \{1,\dots ,n\}\}, \) and the unique minimum element being \( \bigcap \limits _{i=1}^n{\text {null}}(D_i). \) A pseudo-maximal element in \({\text {W}}(D_1,\dots ,D_n)\) is defined as an element of the form

$$\begin{aligned} {\text {null}}(D_i)\cap {\text {null}}(D_j)\ \text {for some }1\le i<j\le n. \end{aligned}$$

Given a tensor \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\), recall \({\mathcal {A}}^\mathsf {T}\) is the tensor by transposing the first and the second indices. Recall that \( D_i:=\big ({\mathcal {A}}^\mathsf {T}\big )_{i, \cdot , \cdot }\) for \(i\in \{1,\dots ,n\} \) are the slices of \({\mathcal {A}}^\mathsf {T}\).

Lemma 4.1

Let \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) and \(B\in {\mathbb {R}}^{n\times n}\). Under either of the following conditions, the mapping \(F(\mathbf{x})={\mathcal {A}}\mathbf{x}^2+B\mathbf{x}+\mathbf c\) is strictly monotone on \({\mathbb {R}}^n_+\) for an arbitrary \(\mathbf c\in {\mathbb {R}}^n\):

(a) :

\({\mathcal {A}}\) is C-strictly copositive for a nonempty closed cone \(C\subseteq {\mathbb {R}}^n_+\), \(B\in \mathbb {R}^{n\times n}\) a positive semidefinite matrix which is positive definite on a nonempty cone \(P\subseteq {\mathbb {R}}^n\) such that \({\mathbb {R}}^n_+\subseteq P\cup C\), and the matrices \(D_1,\dots ,D_n\) are positive semidefinite with

$$\begin{aligned} {\text {null}}(B)\cap W=\{\mathbf 0\} \end{aligned}$$

for every pseudo-maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\);

(b) :

the matrices \(D_1,\dots ,D_n\) and B are positive semidefinite with

$$\begin{aligned} {\text {null}}(B)\cap W=\{\mathbf 0\} \end{aligned}$$

for every maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\).

Proof

We have for any \(\mathbf x,\mathbf y\in {\mathbb {R}}^n_+\)

$$\begin{aligned} \langle F(\mathbf{x})-F(\mathbf{y}),\mathbf{x}-\mathbf{y}\rangle&=\langle {\mathcal {A}}\mathbf{x}^2-{\mathcal {A}}\mathbf{y}^2,\mathbf{x}-\mathbf{y}\rangle +\langle B(\mathbf{x}-\mathbf{y}),\mathbf{x}-\mathbf{y}\rangle \nonumber \\&=\langle {\mathcal {A}}^\mathsf {T}(\mathbf{x}+\mathbf{y})+B,(\mathbf{x}-\mathbf{y})(\mathbf{x}-\mathbf{y})^\mathsf {T}\rangle . \end{aligned}$$
(25)

It is easy to see that under either hypothesis, the tensor \({\mathcal {A}}\) is copositive.

Suppose, without loss of generality, that \(\mathbf y=\mathbf 0\) at first. Then, (25) becomes

$$\begin{aligned} \langle {\mathcal {A}}^\mathsf {T}\mathbf{x}+B,\mathbf{x}\mathbf{x}^\mathsf {T}\rangle . \end{aligned}$$
(26)

If the hypothesis (a) is satisfied, then (26) is nonnegative since \({\mathcal {A}}\) is copositive and B is positive semidefinite. If \(\mathbf{x}^\mathsf {T}B\mathbf{x}=0\), then \(\mathbf x\notin P\), and hence \(\mathbf{x}\in C\), which further implies \(\langle {\mathcal {A}}^\mathsf {T}\mathbf{x},\mathbf{x}\mathbf{x}^\mathsf {T}\rangle >0\). If the other hypothesis (b) is satisfied and \(\mathbf{x}^\mathsf {T}B\mathbf{x}=0\), then \(\mathbf{x}\in {\text {null}}(B)\). Since \(\mathbf{x}\ne \mathbf 0\) and \({\text {null}}(B)\cap W=\{\mathbf 0\}\) for every maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\), it follows that \(\langle {\mathcal {A}}^\mathsf {T}\mathbf{x},\mathbf{x}\mathbf{x}^\mathsf {T}\rangle >0\).

In the following, we suppose that both \(\mathbf{x}\ne \mathbf 0\) and \(\mathbf y\ne \mathbf 0\), and at least two elements of \(\mathbf{x}+\mathbf{y}\) are nonzero, since the case when \(\mathbf{x}+\mathbf{y}\) has only one nonzero component can be proved similarly as the previous argument. Then, at least two matrices \(D_i\)s are involved in \({\mathcal {A}}^\mathsf {T}(\mathbf{x}+\mathbf{y})\). Thus, the null of the matrix \({\mathcal {A}}^\mathsf {T}(\mathbf{x}+\mathbf{y})\) is contained in a pseudo-maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\). Since \(\mathbf{x}\ne \mathbf{y}\) and \({\text {null}}(B)\cap W=\{\mathbf 0\}\) for every pseudo-maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\) under either hypothesis, (25) is positive.\(\square \)

The following proposition can be derived from [2, Theorem 2.3.3].

Proposition 4.1

If \(F(\mathbf{x})={\mathcal {A}}\mathbf{x}^2+B\mathbf{x}+\mathbf c\) is strictly monotone on \({\mathbb {R}}^n_+\), then the solution set \(\textsc {sol}({\mathcal {A}},B,\mathbf c)\) of QCP (1) has at most one element.

The next theorem on the uniqueness follows from Propositions 3.3 and 4.1 and Lemma 4.1.

Theorem 4.1

Let \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^n,3)\) and \(B\in {\mathbb {R}}^{n\times n}\). Suppose that \({\mathcal {A}}\) is C-strictly copositive for a nonempty closed cone \(C\subseteq {\mathbb {R}}^n_+\), B a K-positive semidefinite plus matrix for a nonempty closed cone \(K\subseteq {\mathbb {R}}^n\). Let the intersection of the kernel of B and the linear subspace \({\text {lin}}(K)\) generated by K be \(L\subseteq \mathbb {R}^n\). Suppose that \(K^\complement \cap {\mathbb {R}}^n_+\subseteq C\) and

$$\begin{aligned} L\cap {\mathbb {R}}^n_+\subseteq C\cup \bigg [{\text {int}}\big (({\mathbb {R}}_+\mathbf c)^\diamond \big )\cap {\mathbb {R}}^n_+\bigg ]. \end{aligned}$$

Then, under either of the following conditions:

(a) :

\(B\in \mathbb {R}^{n\times n}\) a positive semidefinite matrix which is positive definite on a cone \(P\subseteq {\mathbb {R}}^n\) such that \({\mathbb {R}}^n_+\subseteq P\cup C\), and the matrices \(D_1,\dots ,D_n\) are positive semidefinite with

$$\begin{aligned} {\text {null}}(B)\cap W=\{\mathbf 0\} \end{aligned}$$

for every pseudo-maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\);

(b) :

the matrices \(D_1,\dots ,D_n\) and B are positive semidefinite with

$$\begin{aligned} {\text {null}}(B)\cap W=\{\mathbf 0\} \end{aligned}$$

for every maximal element \(W\in {\text {W}}(D_1,\dots ,D_n)\);

the QCP (1) has a unique solution.

The next example utilizes the second hypothesis in Theorem 4.1.

Example 4.1

Let \({\mathcal {A}}\in {\text {T}}({\mathbb {R}}^2,3)\) and \(a_{111}=a_{121}=1\) and \(a_{i_1i_2i_3}=0\) for all the other \(i_1, i_2, i_3\in \{1, 2\}\), \(B=\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\). From the given data, \(D_1= D_2=\begin{bmatrix} 1&0\\ 0&0\end{bmatrix}\).

It is easy to verify that \(D_{1}, D_{2}, B\) are positive semidefinite. Let

$$\begin{aligned} C=\{\mathbf x\in {\mathbb {R}}^2_+ : x_1\ge x_2\}. \end{aligned}$$

Then, \(\mathcal A\) is C-strictly copositive and B is \(K={\mathbb {R}}^2_+\)-positive semidefinite plus. Similar as Example 3.3, the regularity condition holds for any \(\mathbf c\in {\mathbb {R}}^2\). It is easy to see that any maximal element of \({\text {W}}(D_1,D_2)\) is \(\{\mathbf x\in {\mathbb {R}}^2 : x_{1}=0\}\), which intersects \({\text {null}}(B)\) trivially. Therefore, the corresponding QCP has a unique solution.

We shall show the uniqueness by direct calculation. The system is

$$\begin{aligned} 0\le x_1^2+x_1x_2+c_1\perp x_1\ge 0\ \text {and } 0\le x_2+c_2\perp x_2\ge 0. \end{aligned}$$

If \(c_2\ge 0\), then \(x_2=0\). Consequently, \(x_1=0\) when \(c_1\ge 0\); and \(x_1=\sqrt{-c_1}\) when \(c_1<0\). If \(c_2<0\), then \(x_2=-c_2\). Consequently, \(x_1=0\) when \(c_1\ge 0\); and \(x_1=\frac{c_2+\sqrt{c_2^2-4c_1}}{2}\) otherwise. Thus, we have uniqueness for each case.

The next example is a modification of Example 4.1 in which the first hypothesis in Theorem 4.1 is conducted.

Example 4.2

Let \({\mathcal {A}}\in {\text {T}}(\mathbb {R}^2, 3)\) and \(a_{111}=1\) and \(a_{i_1i_2i_3}=0\) for all the other \(i_1, i_2, i_3\in \{1, 2\}\). Then, \(D_1=\begin{bmatrix} 1&0\\ 0&0\end{bmatrix}\) and \(D_2=0\). Let \(B=\begin{bmatrix} 0&0\\ 0&1\end{bmatrix}\). We can take \(P=\{\mathbf x\in {\mathbb {R}}^2 : x_2\ne 0\}\). All the other settings are similar to the previous example. The unique pseudo-maximal element in \({\text {W}}(D_1,D_2)\) is

$$\begin{aligned} \{\mathbf x\in {\mathbb {R}}^2: x_{1}=0\}. \end{aligned}$$

All the hypotheses in Theorem 4.1 are satisfied then. Likewise, the uniqueness follows.

5 Conclusions

In this article, we studied existence, compactness and uniqueness of the solution sets of QCPs. Assumptions to guarantee these results are mostly presented in terms of matrices, which should be more tractable. Interestingly, the results in this article generalize the well-known ones in the literature and even broaden the boundary of known knowledge (e.g., Sects. 3.2, 4). These demonstrate that research on QCPs shall be interesting and meaningful for both QCPs and general NCPs. In particular, the study on QCPs would provide fruitful insights on investigations for NCPs.

We conclude this article with remarks that the proposed C-strictly copositivity of a tensor and K-positive semidefiniteness plus of a matrix can be applied to the generalized Markowitz portfolio problem. Actually, the matrix in the QCP reformulation of its optimality condition is K-positive semidefinite plus over the kernel K of that matrix; and the tensor is C-strictly copositive for a proper subcone of the nonnegative orthant. Details and further investigations will be carried out in the coming study.