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Annals of Operations Research

, Volume 265, Issue 1, pp 155–182 | Cite as

LP-based tractable subcones of the semidefinite plus nonnegative cone

  • Akihiro Tanaka
  • Akiko YoshiseEmail author
Open Access
Paths, Pivots, and Practice: The Power of Optimization
  • 703 Downloads

Abstract

The authors in a previous paper devised certain subcones of the semidefinite plus nonnegative cone and showed that satisfaction of the requirements for membership of those subcones can be detected by solving linear optimization problems (LPs) with O(n) variables and \(O(n^2)\) constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this paper, they investigate the properties of the subcones in more detail and explore larger subcones of the positive semidefinite plus nonnegative cone whose satisfaction of the requirements for membership can be detected by solving LPs. They introduce a semidefinite basis (SD basis) that is a basis of the space of \(n \times n\) symmetric matrices consisting of \(n(n+1)/2\) symmetric semidefinite matrices. Using the SD basis, they devise two new subcones for which detection can be done by solving LPs with \(O(n^2)\) variables and \(O(n^2)\) constraints. The new subcones are larger than the ones in the previous paper and inherit their nice properties. The authors also examine the efficiency of those subcones in numerical experiments. The results show that the subcones are promising for testing copositivity as a useful application.

Keywords

Semidefinite plus nonnegative cone Doubly nonnegative cone Copositive cone Matrix decomposition Linear programming Semidefinite basis Maximum clique problem Quadratic optimization problem 

1 Introduction

Let \(\mathcal {S}_n\) be the set of \(n \times n\) symmetric matrices, and define their inner product as
$$\begin{aligned} \langle A, B \rangle = \mathrm{Tr}\,(B^TA) = \sum _{i.j=1}^n a_{ij}b_{ij}. \end{aligned}$$
(1)
Bomze et al. (2000) coined the term “copositive programming” in relation to the following problem in 2000, on which many studies have since been conducted:
$$\begin{aligned} \begin{array}{ll} \text{ Minimize } &{} \langle C, X \rangle \\ \text{ subject } \text{ to } &{} \langle A_i, X \rangle = b_i, \ (i=1,2,\ldots ,m) \\ &{} X \in \mathcal{COP}_n. \end{array} \end{aligned}$$
where \(\mathcal {COP}_n\) is the set of \(n \times n\) copositive matrices, i.e., matrices whose quadratic form takes nonnegative values on the n-dimensional nonnegative orthant \(\mathbb {R}^n_+\):
$$\begin{aligned} \mathcal {COP}_n := \{ X \in \mathcal {S}_n \mid d^T Xd \ge 0 \ \text{ for } \text{ all } \ d \in \mathbb {R}^n_+ \}. \end{aligned}$$
We call the set \(\mathcal {COP}_n\) the copositive cone. A number of studies have focused on the close relationship between copositive programming and quadratic or combinatorial optimization (see, e.g., Bomze et al. 2000; Bomze and Klerk 2002; Klerk and Pasechnik 2002; Povh and Rendl 2007, 2009; Bundfuss 2009; Burer 2009; Dickinson and Gijben 2014). Interested readers may refer to Dür (2010) and Bomze (2012) for background on and the history of copositive programming.
The following cones are attracting attention in the context of the relationship between combinatorial optimization and copositive optimization (see, e.g., Dür 2010; Bomze 2012). Here, \(\mathrm{conv}\,(S)\) denotes the convex hull of the set S.
  • The nonnegative cone \(\mathcal{N}_n := \left\{ X \in \mathcal {S}_n \mid x_{ij} \ge 0 \text{ for } \text{ all } \ i, j \in \{1,2,\ldots ,n\} \right\} \).

  • The semidefinite cone \(\mathcal{S}_n^+ := \{X \in \mathcal {S}_n \mid d^TXd \ge 0 \ \text{ for } \text{ all } \ d \in \mathbb {R}^n \} = \mathrm{conv}\,\left( \left\{ xx^T \mid x \in \mathbb {R}^n \right\} \right) \).

  • The copositive cone \(\mathcal{COP}_n := \left\{ X \in \mathcal {S}_n \mid d^TXd \ge 0 \ \text{ for } \text{ all } \ d \in \mathbb {R}^n_+ \right\} \).

  • The semidefinite plus nonnegative cone \(\mathcal{S}_n^+ + \mathcal{N}_n\), which is the Minkowski sum of \(\mathcal{S}_n^+\) and \(\mathcal{N}_n\).

  • The union \(\mathcal{S}_n^+ \cup \mathcal{N}_n\) of \(\mathcal{S}_n^+\) and \(\mathcal{N}_n\).

  • The doubly nonnegative cone \(\mathcal{S}_n^+ \cap \mathcal{N}_n\), i.e., the set of positive semidefinite and componentwise nonnegative matrices.

  • The completely positive cone \(\mathcal{CP}_n := \mathrm{conv}\,\left( \left\{ xx^T \mid x \in \mathbb {R}^n_+ \right\} \right) \).

Except the set \(\mathcal{S}_n^+ \cup \mathcal{N}_n\), all of the above cones are proper (see Section 1.6 of Berman and Monderer (2003), where a proper cone is called a full cone), and we can easily see from the definitions that the following inclusions hold:
$$\begin{aligned} \mathcal{COP}_n \supseteq \mathcal{S}_n^+ + \mathcal{N}_n \supseteq \mathcal{S}_n^+ \cup \mathcal{N}_n \supseteq \mathcal{S}_n^+ \supseteq \mathcal{S}_n^+ \cap \mathcal{N}_n \supseteq \mathcal{CP}_n. \end{aligned}$$
(2)
While copositive programming has the potential of being a useful optimization technique, it still faces challenges. One of these challenges is to develop efficient algorithms for determining whether a given matrix is copositive. It has been shown that the above problem is co-NP-complete (Murty and Kabadi 1987; Dickinson 2014; Dickinson and Gijben 2014) and many algorithms have been proposed to solve it (see, e.g., Bomze 1996; Bundfuss and Dür 2008; Johnson and Reams 2008; Jarre and Schmallowsky 2009; Z̆ilinskas and Dür 2011; Sponsel et al. 2012; Bomze and Eichfelder 2013; Deng et al. 2013; Dür and Hiriart-Urruty 2013; Tanaka and Yoshise 2015; Brás et al. 2015) Here, we are interested in numerical algorithms which (a) apply to general symmetric matrices without any structural assumptions or dimensional restrictions and (b) are not merely recursive, i.e., do not rely on information taken from all principal submatrices, but rather focus on generating subproblems in a somehow data-driven way, as described in Bomze and Eichfelder (2013). There are few such algorithms, but they often use tractable subcones \(\mathcal {M}_n\) of the semidefinite plus nonnegative cone \(\mathcal{S}_n^+ + \mathcal{N}_n\) for detecting copositivity (see, e.g., Bundfuss and Dür 2008; Sponsel et al. 2012; Bomze and Eichfelder 2013; Tanaka and Yoshise 2015). As described in Sect. 5, these algorithms require one to check whether \(A \in \mathcal {M}_n\) or \(A \not \in \mathcal {M}_n\) repeatedly over simplicial partitions. The desirable properties of the subcones \(\mathcal {M}_n \subseteq \mathcal{S}_n^+ + \mathcal{N}_n\) used by these algorithms can be summarized as follows:
  1. P1

    For any given \(n \times n\) symmetric matrix \(A \in \mathcal {S}_n\), we can check whether \(A \in \mathcal {M}_n\) within a reasonable computation time, and

     
  2. P2

    \(\mathcal {M}_n\) is a subset of the semidefinite plus nonnegative cone \(\mathcal{S}_n^+ + \mathcal{N}_n\) that at least includes the \(n \times n\) nonnegative cone \(\mathcal {N}_n\) and contains as many elements \(\mathcal{S}_n^+ + \mathcal{N}_n\) as possible.

     
The authors, in Tanaka and Yoshise (2015), devised certain subcones of the semidefinite plus nonnegative cone \(\mathcal{S}_n^+ + \mathcal{N}_n\) and showed that satisfaction of the requirements for membership of those cones can be detected by solving linear optimization problems (LPs) with O(n) variables and \(O(n^2)\) constraints. They also created an LP-based algorithm that uses these subcones for testing copositivity as an application of those cones.

The aim of this paper is twofold. First, we investigate the properties of the subcones in more detail, especially in terms of their convex hulls. Second, we search for subcones of the semidefinite plus nonnegative cone \(\mathcal{S}_n^+ + \mathcal{N}_n\) that have properties P1 and P2. To address the second aim, we introduce a semidefinite basis (SD basis) that is a basis of the space \(\mathcal {S}_n\) consisting of \(n(n+1)/2\) symmetric semidefinite matrices. Using the SD basis, we devise two new types of subcones for which detection can be done by solving LPs with \(O(n^2)\) variables and \(O(n^2)\) constraints. As we will show in Corollary 1, these subcones are larger than the ones proposed in Tanaka and Yoshise (2015) and inherit their nice properties. We also examine the efficiency of those subcones in numerical experiments.

This paper is organized as follows: In Sect. 2, we show several tractable subcones of \(\mathcal{S}_n^+ + \mathcal{N}_n\) that are receiving much attention in the field of copositive programming and investigate their properties, the results of which are summarized in Figs. 1 and 2. In Sect. 3, we propose new subcones of \(\mathcal{S}_n^+ + \mathcal{N}_n\) having properties P1 and P2. We define SD bases using Definitions 1 and 2 and construct new LPs for detecting whether a given matrix belongs to the subcones. In Sect. 4, we perform numerical experiments in which the new subcones are used for identifying the given matrices \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\). As a useful application of the new subcones, Sect. 5 describes experiments for testing copositivity of matrices arising from the maximum clique problem and standard quadratic optimization problems. The results of these experiments show that the new subcones are promising not only for identification of \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) but also for testing copositivity. We give concluding remarks in Sect. 6.

2 Some tractable subcones of \(\mathcal{S}_n^+ + \mathcal{N}_n\) and related work

In this section, we show several tractable subcones of the semidefinite plus nonnegative cone \(\mathcal{S}_n^+ + \mathcal{N}_n\). Since the set \(\mathcal{S}_n^+ + \mathcal{N}_n\) is the dual cone of the doubly nonnegative cone \(\mathcal{S}_n^+ \cap \mathcal{N}_n\), we see that
$$\begin{aligned} \mathcal{S}_n^+ + \mathcal{N}_n= & {} \{ A \in \mathcal{S}_n \mid \langle A, X \rangle \ge 0 \ \text{ for } \text{ any } \ X \in \mathcal{S}_n^+ \cap \mathcal{N}_n \} \\= & {} \{ A \in \mathcal{S}_n \mid \langle A, X \rangle \ge 0 \ \text{ for } \text{ any } \ X \in \mathcal{S}_n^+ \cap \mathcal{N}_n \ \text{ such } \text{ that } \ \mathrm{Tr}\,(X) = 1 \} \end{aligned}$$
and that the weak membership problem for \(\mathcal{S}_n^+ + \mathcal{N}_n\) can be solved (to an accuracy of \(\epsilon \)) by solving the following doubly nonnegative program (which can be expressed as a semidefinite program of size \(O(n^2)\)).
$$\begin{aligned} \begin{array}{ll} \text{ Minimize } &{} \langle A, X \rangle \\ \text{ subject } \text{ to } &{} \langle I_n, X \rangle = 1, \ X \in \mathcal{S}_n^+ \cap \mathcal{N}_n \end{array} \end{aligned}$$
(3)
where \(I_n\) denotes the \(n \times n\) identity matrix. Thus, the set \(\mathcal{S}_n^+ + \mathcal{N}_n\) is a rather large and tractable convex subcone of \(\mathcal {COP}_n\). However, solving the problem takes a lot of time (Sponsel et al. 2012; Yoshise and Matsukawa 2010) and does not make for a practical implementation in general. To overcome this drawback, more easily tractable subcones of \(\mathcal{S}_n^+ + \mathcal{N}_n\) have been proposed.
We define the matrix functions \(N, S: \mathcal{S}_n \rightarrow \mathcal{S}_n\) such that, for \(A \in \mathcal {S}_n\), we have
$$\begin{aligned} N(A)_{ij}:= \left\{ \begin{array}{ll} A_{ij} &{} \ \ (A_{ij}>0 \ \text{ and } \ i \ne j) \\ 0 &{} \ \ (\text{ otherwise }) \end{array} \right. \ \ \text{ and } \ \ S(A):= A-N(A). \end{aligned}$$
(4)
In Sponsel et al. (2012), the authors defined the following set:
$$\begin{aligned} \mathcal {H}_n := \{ A \in \mathcal {S}_n \mid S(A) \in \mathcal {S}_n^+ \}. \end{aligned}$$
(5)
Here, we should note that \(A = S(A)+ N(A) \in \mathcal {S}_n^+ + \mathcal {N}_n\) if \(A \in \mathcal {H}_n\). Also, for any \(A \in \mathcal {N}_n\), S(A) is a nonnegative diagonal matrix, and hence, \(\mathcal {N}_n\subseteq \mathcal {H}_n\). The determination of \(A \in \mathcal {H}_n\) is easy and can be done by extracting the positive elements \(A_{ij} > 0 \ (i \ne j)\) as \(N(A)_{ij}\) and by performing a Cholesky factorization of S(A) (cf. Algorithm 4.2.4 in Golub and Van Loan 1996). Thus, from the inclusion relation (2), we see that the set \(\mathcal {H}_n\) has the desirable P1 property. However, S(A) is not necessarily positive semidefinite even if \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) or \(A \in \mathcal {S}_n^+\). The following theorem summarizes the properties of the set \(\mathcal {H}_n\).

Theorem 1

[Fiedler and Pták (1962) and Theorem 4.2 of Sponsel et al. (2012)] \(\mathcal {H}_n\) is a convex cone and \(\mathcal {N}_n \subseteq \mathcal {H}_n \subseteq \mathcal {S}_n^+ + \mathcal {N}_n\). If \(n \ge 3\), these inclusions are strict and \(\mathcal{S}_n^+ \not \subseteq \mathcal {H}_n\). For \(n=2\), we have \(\mathcal {H}_n = \mathcal {S}_n^+ \cup \mathcal {N}_n = \mathcal {S}_n^+ + \mathcal {N}_n =\mathcal {COP}_n\).

The construction of the subcone \(\mathcal {H}_n\) is based on the idea of “checking nonnegativity first and checking positive semidefiniteness second.” In Tanaka and Yoshise (2015), another subcone is provided that is based on the idea of “checking positive semidefiniteness first and checking nonnegativity second.” Let \(\mathcal{O}_n\) be the set of \(n \times n\) orthogonal matrices and \(\mathcal{D}_n\) be the set of \(n \times n\) diagonal matrices. For a given symmetric matrix \(A \in \mathcal {S}_n\), suppose that \(P=[p_1, p_2, \ldots , p_n] \in \mathcal{O}_n\) and \(\varLambda = \mathrm{Diag}\,(\lambda _1, \lambda _2, \ldots , \lambda _n) \in \mathcal{D}_n\) satisfy
$$\begin{aligned} A = P\varLambda P^T = \sum _{i=1}^n \lambda _i p_ip_i^T. \end{aligned}$$
(6)
By introducing another diagonal matrix \(\varOmega = \mathrm{Diag}\,(\omega _1, \omega _2, \ldots , \omega _n) \in \mathcal{D}_n\), we can make the following decomposition:
$$\begin{aligned} A= P(\varLambda -\varOmega )P^T+P\varOmega P^T \end{aligned}$$
(7)
If \(\varLambda -\varOmega \in \mathcal{N}_n\), i.e., if \(\lambda _i \ge \omega _i \ (i = 1,2, \ldots ,n)\), then the matrix \(P(\varLambda -\varOmega )P^T\) is positive semidefinite. Thus, if we can find a suitable diagonal matrix \(\varOmega \in \mathcal{D}_n\) satisfying
$$\begin{aligned} \lambda _i \ge \omega _i \ (i = 1,2, \ldots ,n), \ \ [P\varOmega P^T]_{ij} \ge 0 \ (1 \le i \le j \le n) \end{aligned}$$
(8)
then (7) and (2) imply
$$\begin{aligned} A= P(\varLambda -\varOmega )P^T+P\varOmega P^T \in \mathcal {S}_n^+ + \mathcal {N}_n \subseteq \mathcal {COP}_n. \end{aligned}$$
(9)
We can determine whether such a matrix exists or not by solving the following linear optimization problem with variables \(\omega _i \ (i=1,2,\ldots ,n) \) and \(\alpha \):
$$\begin{aligned} \text{(LP) }_{P,\varLambda } \ \left| \begin{array}{lll} \text{ Maximize } &{} \alpha &{} \\ \text{ subject } \text{ to } &{} \omega _i \le \lambda _i &{} (i=1,2,\ldots ,n) \\ &{} \displaystyle { [P\varOmega P^T]_{ij} = \left[ \sum _{k=1}^n \omega _k p_kp_k^T \right] _{ij} \ge \alpha } &{} (1 \le i \le j \le n) \end{array} \right. \end{aligned}$$
(10)
Here, for a given matrix A, \([A]_{ij}\) denotes the (ij)th element of A.
Problem \(\text{(LP) }_{P,\varLambda }\) has a feasible solution at which \(\omega _i = \lambda _i \ (i=1,2,\ldots ,n)\) and
$$\begin{aligned} \alpha = \min \left\{ \left[ P\varLambda P^T \right] _{ij} \mid 1 \le i \le j \le n \right\} = \min \left\{ \sum _{k=1}^n \lambda _k [p_k]_i[p_k]_j \mid 1 \le i \le j \le n \right\} . \end{aligned}$$
For each \(i=1,2,\ldots ,n\), the constraints
$$\begin{aligned}{}[P\varOmega P^T]_{ii} = \left[ \sum _{k=1}^n \omega _k p_kp_k^T \right] _{ii} = \sum _{k=1}^n \omega _k [p_k]_i^2 \ge \alpha \end{aligned}$$
and \(\omega _k \le \lambda _k \ (k=1,2,\ldots ,n)\) imply the bound \(\alpha \le \min \left\{ \sum _{k=1}^n \lambda _k [p_k]_i^2 \mid 1 \le i \le n \right\} \). Thus, \(\text{(LP) }_{P,\varLambda }\) has an optimal solution with optimal value \(\alpha _*(P, \varLambda )\). If \(\alpha _*(P, \varLambda ) \ge 0\), there exists a matrix \(\varOmega \) for which the decomposition (8) holds. The following set \(\mathcal {G}_n\) is based on the above observations and was proposed in Tanaka and Yoshise (2015) as the set, \(\mathcal {G}_n\)
$$\begin{aligned} \mathcal {G}_n :=\{A \in \mathcal {S}_n \mid {\mathcal {P}}{\mathcal {L}}_{\mathcal {G}_n}(A) \ne \emptyset \} \end{aligned}$$
(11)
where
$$\begin{aligned} {\mathcal {P}}{\mathcal {L}}_{\mathcal {G}_n}(A) := \{ (P, \varLambda ) \in \mathcal{O}_n \times \mathcal{D}_n \mid P \text{ and } \varLambda \text{ satisfy } (6) \text{ and } \alpha _*(P,\varLambda ) \ge 0 \} \end{aligned}$$
(12)
for a given \(A \in \mathcal{S}_n\). As stated above, if \(\alpha _*(P,\varLambda ) \ge 0\) for a given decomposition \(A = P\varLambda P^T\), we can determine \(A \in \mathcal {G}_n\). In this case, we just need to compute a matrix decomposition and solve a linear optimization problem with \(n+1\) variables and \(O(n^2)\) constraints, which implies that it is rather practical to use the set \(\mathcal {G}_n\) as an alternative to using \(\mathcal{S}_n^+ + \mathcal{N}_n\). Suppose that \(A \in \mathcal {S}_n\) has n different eigenvalues. Then the possible orthogonal matrices \(P = [p_1, p_2, \ldots , p_n] \in \mathcal{O}_n\) are identifiable, except for the permutation and sign inversion of \(\{p_1, p_2, \ldots , p_n \}\), and by representing (6) as
$$\begin{aligned} A = \sum _{i=1}^n \lambda _i p_ip_i^T, \end{aligned}$$
we can see that the problem \(\text{(LP) }_{P,\varLambda }\) is unique for any possible \(P \in \mathcal{O}_n\). In this case, \(\alpha _*(P,\varLambda ) < 0\) with a specific \(P \in \mathcal{O}_n\) implies \(A \not \in \mathcal {G}_n\). However, if this is not the case (i.e., an eigenspace of A has at least dimension 2), \(\alpha _*(P,\varLambda ) < 0\) with a specific \(P \in \mathcal{O}_n\) does not necessarily guarantee that \(A \not \in \mathcal {G}_n\).

The above discussion can be extended to any matrix \(P \in \mathbb {R}^{m \times n}\); i.e., it does not necessarily have to be orthogonal or even square. The reason why the orthogonal matrices \(P \in \mathcal{O}_n\) are dealt with here is that some decomposition methods for (6) have been established for such orthogonal Ps. The property \(\mathcal {G}_n = \text{ com }(\mathcal {S}_n, \mathcal {N}_n)\) in Theorem 2 also follows when P is orthogonal.

In Tanaka and Yoshise (2015), the authors described another set \(\widehat{\mathcal {G}_n}\) that is closely related to \(\mathcal {G}_n\).
$$\begin{aligned} \widehat{\mathcal {G}_n} := \{A \in \mathcal {S}_n \mid {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {G}_n}}(A) \ne \emptyset \} \end{aligned}$$
(13)
where for \(A \in \mathcal{S}_n\), the set \({\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {G}_n}}(A)\) is given by replacing \(\mathcal{O}_n\) in (12) by the space \(\mathbb {R}^{n \times n}\) of \(n \times n\) arbitrary matrices, i.e.,
$$\begin{aligned} {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {G}_n}}(A) := \{ (P, \varLambda ) \in \mathbb {R}^{n\times n} \times \mathcal{D}_n \mid P \text{ and } \varLambda \text{ satisfy } (6) \text{ and } \alpha _*(P,\varLambda ) \ge 0 \}. \end{aligned}$$
(14)
If the set \({\mathcal {P}}{\mathcal {L}}_{\mathcal {G}_n}(A)\) in (12) is nonempty, then the set \({\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {G}_n}}(A)\) is also nonempty, which implies the following inclusions:
$$\begin{aligned} \mathcal {G}_n \subseteq \widehat{\mathcal {G}_n} \subseteq \mathcal {S}_n^+ + \mathcal {N}_n. \end{aligned}$$
(15)
Before describing the properties of the sets \(\mathcal {G}_n\) and \(\widehat{\mathcal {G}_n}\), we will prove a preliminary lemma.

Lemma 1

Let \(\mathcal {K}_1\) and \(\mathcal {K}_2\) be two convex cones containing the origin. Then \(\mathrm{conv}\,(\mathcal {K}_1 \cup \mathcal {K}_2) = \mathcal {K}_1 + \mathcal {K}_2\).

Proof

Since \(\mathcal {K}_1\) and \(\mathcal {K}_2\) are convex cones, we can easily see that the inclusion \(\mathcal {K}_1 + \mathcal {K}_2 \subseteq \mathrm{conv}\,(\mathcal {K}_1 \cup \mathcal {K}_2)\) holds. The converse inclusion also follows from the fact that \(\mathcal {K}_1\) and \(\mathcal {K}_2\) are convex cones. Since \(\mathcal {K}_1\) and \(\mathcal {K}_2\) contain the origin, we see that the inclusion \(\mathcal {K}_1 \cup \mathcal {K}_2 \subseteq \mathcal {K}_1 + \mathcal {K}_2\) holds. From this inclusion and the convexity of the sets \(\mathcal {K}_1\) and \(\mathcal {K}_2\), we can conclude that
$$\begin{aligned} \mathrm{conv}\,(\mathcal {K}_1 \cup \mathcal {K}_2) \subseteq \mathrm{conv}\,(\mathcal {K}_1 + \mathcal {K}_2) = \mathcal {K}_1 + \mathcal {K}_2. \end{aligned}$$
\(\square \)

The following theorem shows some of the properties of \(\mathcal {G}_n\) and \(\widehat{\mathcal {G}_n}\). Assertions (i) and (ii) were proved in Theorem 3.2 of Tanaka and Yoshise (2015). Assertion (iii) comes from the fact that \(\mathcal {S}_n^+\) and \(\mathcal {N}_n\) are convex cones and from Lemma 1. Assertions (iv)–(vi) follow from (i)–(iii), the inclusion (15) and Theorem 1.

Theorem 2

  1. (i)

    \(\mathcal {S}_n^+ \cup \mathcal {N}_n \subseteq \mathcal {G}_n\)

     
  2. (ii)
    \(\mathcal {G}_n = \mathrm{com}\,(\mathcal {S}_n^+, \mathcal {N}_n)\), where the set \(\mathrm{com}\,(\mathcal {S}_n^+, \mathcal {N}_n)\) is defined by
    $$\begin{aligned} \mathrm{com}\,(\mathcal {S}_n^+, \mathcal {N}_n) := \{ S + N \mid S \in \mathcal {S}_n^+, \ N \in \mathcal {N}_n, \ \text{ S } \text{ and } \text{ N } \text{ commute } \}. \end{aligned}$$
     
  3. (iii)

    \(\mathrm{conv}\,(\mathcal {S}_n^+ \cup \mathcal {N}_n) = \mathcal {S}_n^+ + \mathcal {N}_n\).

     
  4. (iv)

    \( \mathcal {S}_n^+ \cup \mathcal {N}_n \subseteq \mathcal {G}_n = \text{ com }(\mathcal {S}_n^+, \mathcal {N}_n) \subseteq \widehat{\mathcal {G}_n} \subseteq \mathcal {S}_n^+ + \mathcal {N}_n \).

     
  5. (v)

    If \(n = 2\), then \( \mathcal {S}_n^+ \cup \mathcal {N}_n = \mathcal {G}_n = \mathrm{com}\,(\mathcal {S}_n^+, \mathcal {N}_n) = \widehat{\mathcal {G}_n} = \mathcal {S}_n^+ + \mathcal {N}_n \)

     
  6. (vi)

    \( \mathrm{conv}\,(\mathcal {S}_n^+ \cup \mathcal {N}_n) = \mathrm{conv}\,(\mathcal {G}_n) = \mathrm{conv}\,\left( \mathrm{com}\,(\mathcal {S}_n^+, \mathcal {N}_n)\right) = \mathrm{conv}\,(\widehat{\mathcal {G}_n}) = \mathcal {S}_n^+ + \mathcal {N}_n \).

     
A number of examples provided in Tanaka and Yoshise (2015) illustrate the differences between \(\mathcal {H}_n\), \(\mathcal {G}_n\). Moreover, the following two matrices have three different eigenvalues, respectively, and we can identify
$$\begin{aligned} \left[ \begin{array}{r@{\quad }r@{\quad }r} 2 &{} 2 &{} 2\\ 2 &{} 2 &{} -\,3\\ 2 &{} -\,3 &{} 6 \end{array} \right] \in \mathcal {H}_3 {\setminus } \mathcal {G}_3, \ \ \left[ \begin{array}{r@{\quad }r@{\quad }r} 1 &{} 5 &{} -\,2\\ 5 &{} 1 &{} -\,2 \\ -\,2 &{} -\,2 &{} 4 \end{array} \right] \in (\mathcal {S}_3^+ + \mathcal {N}_3) {\setminus } (\mathcal {H}_3 \cup \mathcal {G}_3) \end{aligned}$$
(16)
by solving the associated LPs. Figure 1 draws those examples and (ii) of Theorem 2. Figure 2 follows from (vii) of Theorem 2 and the convexity of the sets \(\mathcal {N}_n\), \(\mathcal {S}_n^+\) and \(\mathcal {H}_n\) (see Theorem 1).
Fig. 1

Examples of inclusion relations among the subcones of \(\mathcal {S}_n^+ + \mathcal {N}_n\) I

Fig. 2

Examples of inclusion relations among the subcones of \(\mathcal {S}_n^+ + \mathcal {N}_n\) II

At present, it is not clear whether the set \(\mathcal {G}_n = \mathrm{com}\,(\mathcal {S}_n^+, \mathcal {N}_n)\) is convex or not. As we will mention our numerical results suggest that the set might be not convex.

Before closing this discussion, we should point out another interesting subset of \(\mathcal {S}_n^+ + \mathcal {N}_n\) proposed by Bomze and Eichfelder (2013). Suppose that a given matrix \(A \in \mathcal {S}_n\) can be decomposed as (6), and define the diagonal matrix \(\varLambda _+\) by \( [\varLambda _+]_{ii} = \max \{0, \lambda _i \}\). Let \(A_+: = P\varLambda _+P^T\) and \(A_- := A_+ - A\). Then, we can easily see that \(A_+\) and \(A_-\) are positive semidefinite. Using this decomposition \(A = A_+ - A_-\), Bomze and Eichfelder derived the following LP-based sufficient condition for \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) in Bomze and Eichfelder (2013).

Theorem 3

[Theorem 2.6 of Bomze and Eichfelder (2013)] Let \(x \in \mathbb {R}_n^+\) be such that \(A_+ x\) has only positive coordinates. If
$$\begin{aligned} (x^TA_+ x) (A_-)_{ii} \le [(A_+x)_i ]^2 \ (i=1,2,\ldots ,n) \end{aligned}$$
then \(A \in \mathcal {COP}_n\).
Consider the following LP with O(n) variables and O(n) constraints,
$$\begin{aligned} \inf \{ f^Tx \mid A_+ x \ge e, \ x \in \mathbb {R}_n^+ \} \end{aligned}$$
(17)
where f is an arbitrary vector and e denotes the vector of all ones. Define the set,
$$\begin{aligned} \mathcal {L}_n:= & {} \{ A \in \mathcal {S}_n \mid (x^TA_+ x) (A_-)_{ii} \le [(A_+x)_i ]^2 \ (i=1,2,\ldots ,n) \ \text{ for } \text{ some } \text{ feasible } \text{ solution } \\&x \text{ of } (17) \}. \end{aligned}$$
Then Theorem 3 ensures that \(\mathcal {L}_n \subseteq \mathcal {COP}_n\). The following proposition gives a characterization when the feasible set of the LP of (17) is empty.

Proposition 1

[Proposition 2.7 of Bomze and Eichfelder (2013)] The condition \(\mathrm{ker} A_+ \cap \{x \in \mathbb {R}_n^+ \mid e^Tx =1 \} \ne \emptyset \) is equivalent to \( \{x \in \mathbb {R}_n^+ \mid A_+ x \ge e\} = \emptyset \).

Consider the matrix,
$$\begin{aligned} A = \left[ \begin{array}{rr} 1&{}\quad -\,1\\ -\,1&{}\quad 1\\ \end{array} \right] \in \mathcal {S}_2^+. \end{aligned}$$
Thus, \(A_+ = A\), and the set \(\mathrm{ker} A_+ \cap \{x \in \mathbb {R}_n^+ \mid e^Tx =1 \} \ne \emptyset \). Proposition 1 ensures that \(A \not \in \mathcal {L}_2\), and hence, \(\mathcal {S}_n^+ \not \subseteq \mathcal {L}_n\) for \(n \ge 2\), similarly to the set \(\mathcal {H}_n\) for \(n \ge 3\) (see Theorem 1).

3 Semidefinite bases

In this section, we improve the subcone \(\mathcal {G}_n\) in terms of P2. For a given matrix A of (6), the linear optimization problem \(\text{(LP) }_{P,\varLambda }\) in (10) can be solved in order to find a nonnegative matrix that is a linear combination
$$\begin{aligned} \sum _{i=1}^n \omega _i p_ip_i^T \end{aligned}$$
of n linearly independent positive semidefinite matrices \(p_ip_i^T \in \mathcal {S}_n^+ \ (i=1,2,\ldots ,n)\). This is done by decomposing \(A \in \mathcal {S}_n\) into two parts:
$$\begin{aligned} A = \sum _{i=1}^n (\lambda _i - \omega _i) p_ip_i^T + \sum _{i=1}^n \omega _i p_ip_i^T \end{aligned}$$
(18)
such that the first part
$$\begin{aligned} \sum _{i=1}^n (\lambda _i - \omega _i) p_ip_i^T \end{aligned}$$
is positive semidefinite. Since \(p_ip_i^T \in \mathcal {S}_n^+ \ (i=1,2,\ldots ,n)\) are only n linearly independent matrices in \(n(n+1)/2\) dimensional space \(\mathcal {S}_n\), the intersection of the set of linear combinations of \(p_ip_i^T\) and the nonnegative cone \(\mathcal {N}_n\) may not have a nonzero volume even if it is nonempty. On the other hand, if we have a set of positive semidefinite matrices \(p_ip_i^T \in \mathcal {S}_n^+ \ (i=1,2,\ldots ,n(n+1)/2)\) that gives a basis of \(\mathcal {S}_n\), then the corresponding intersection becomes the nonnegative cone \(\mathcal {N}_n\) itself, and we may expect a greater chance of finding a nonnegative matrix by enlarging the feasible region of \(\text{(LP) }_{P,\varLambda }\). In fact, we can easily find a basis of \(\mathcal {S}_n\) consisting of \(n(n+1)/2\) semidefinite matrices from n given orthogonal vectors \(p_i \in \mathbb {R}^n \ (i=1,2,\ldots ,n)\) based on the following result from Dickinson (2011).

Proposition 2

[Lemma 6.2 of Dickinson (2011)] Let \(v_i \in \mathbb {R}^n (i=1,2,\ldots ,n)\) be n-dimensional linear independent vectors. Then the set \(\mathcal {V} := \{(v_i + v_j)(v_i + v_j)^T \mid 1 \le i \le j \le n\}\) is a set of \(n(n+1)/2\) linearly independent positive semidefinite matrices. Therefore, the set \(\mathcal {V}\) gives a basis of the set \(\mathcal {S}_n\) of \(n \times n\) symmetric matrices.

The above proposition ensures that the following set \(\mathcal {B}_+(p_1,p_2,\ldots ,p_n)\) is a basis of \(n \times n\) symmetric matrices.

Definition 1

(Semidefinite basis type I) For a given set of n-dimensional orthogonal vectors \(p_i \in \mathbb {R}^n (i=1,2,\ldots ,n)\), define the map \(\varPi _+: \mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathcal {S}_n^+\) by
$$\begin{aligned} \varPi _+(p_i, p_j) := \frac{1}{4}(p_i + p_j)(p_i + p_j)^T. \end{aligned}$$
(19)
We call the set
$$\begin{aligned} \mathcal {B}_+(p_1,p_2,\ldots ,p_n) := \{ \varPi _+(p_i,p_j) \mid 1 \le i \le j \le n \} \end{aligned}$$
(20)
a semidefinite basis type I induced by \(p_i \in \mathbb {R}^n (i=1,2,\ldots ,n)\).
A variant of the semidefinite basis type I is as follows. Noting that the equivalence
$$\begin{aligned} \varPi _+(p_i,p_j) = \frac{1}{2}p_ip_i^T+ \frac{1}{2}p_jp_j^T - \varPi _-(p_i,p_j) \end{aligned}$$
holds for any \(i \ne j\), we see that \(\mathcal {B}_-(p_1,p_2,\ldots ,p_n)\) is also a basis of \(n \times n\) symmetric matrices.

Definition 2

(Semidefinite basis type II) For a given set of n-dimensional orthogonal vectors \(p_i \in \mathbb {R}^n (i=1,2,\ldots ,n)\), define the map \(\varPi _+: \mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathcal {S}_n^+\) by
$$\begin{aligned} \varPi _-(p_i, p_j) := \frac{1}{4}(p_i - p_j)(p_i - p_j)^T. \end{aligned}$$
(21)
We call the set
$$\begin{aligned} \mathcal {B}_-(p_1,p_2,\cdots ,p_n) := \{ \varPi _+(p_i,p_i) \mid 1 \le i \le n \} \cup \{ \varPi _-(p_i,p_j) \mid 1 \le i < j \le n \}\qquad \end{aligned}$$
(22)
a semidefinite basis type II induced by \(p_i \in \mathbb {R}^n (i=1,2,\ldots ,n)\).
Using the map \(\varPi _+\) in (19), the linear optimization problem \(\text{(LP) }_{P,\varLambda }\) in (10) can be equivalently written as
$$\begin{aligned} \text{(LP) }_{P,\varLambda } \ \left| \begin{array}{lll} \text{ Maximize } &{} \alpha &{} \\ \text{ subject } \text{ to } &{} \omega _{ii}^+ \le \lambda _i &{} (i=1,2,\ldots ,n) \\ &{} \displaystyle { \left[ \sum _{k=1}^n \omega _{kk}^+ \varPi _+(p_k,p_k) \right] _{ij} \ge \alpha } &{} (1 \le i \le j \le n). \end{array} \right. \end{aligned}$$
The problem \(\text{(LP) }_{P,\varLambda }\) is based on the decomposition (18). Starting with (18), the matrix A can be decomposed using \(\varPi _+(p_i,p_j)\) in (19) and \(\varPi _-(p_i,p_j)\) in (21) as
$$\begin{aligned} A= & {} \sum _{i=1}^n (\lambda _i-\omega ^+_{ii}) \varPi _+(p_i, p_i) + \sum _{i=1}^n \omega ^+_{ii} \varPi _+(p_i, p_i) \nonumber \\= & {} \sum _{i=1}^n (\lambda _i-\omega ^+_{ii}) \varPi _+(p_i, p_i) + \sum _{i=1}^n \omega ^+_{ii} \varPi _+(p_i, p_i) \nonumber \\&+ \sum _{1 \le i< j \le n} (- \omega ^+_{ij}) \varPi _+(p_i, p_j) + \sum _{1 \le i < j \le n} \omega ^+_{ij} \varPi _+(p_i, p_j) \end{aligned}$$
(23)
$$\begin{aligned}= & {} \sum _{i=1}^n (\lambda _i-\omega ^+_{ii}) \varPi _+(p_i, p_i) + \sum _{i=1}^n \omega ^+_{ii} \varPi _+(p_i, p_i) \nonumber \\&+ \sum _{1 \le i< j \le n} (- \omega ^+_{ij}) \varPi _+(p_i, p_j) + \sum _{1 \le i< j \le n} \omega ^+_{ij} \varPi _+(p_i, p_j) \nonumber \\&+ \sum _{1 \le i< j \le n} (- \omega ^-_{ij}) \varPi _-(p_i, p_j) + \sum _{1 \le i < j \le n} \omega ^-_{ij} \varPi _-(p_i, p_j) . \end{aligned}$$
(24)
On the basis of the decomposition (23) and (24), we devise the following two linear optimization problems as extensions of \(\text{(LP) }_{P,\varLambda }\):
$$\begin{aligned}&\text{(LP) }_{P,\varLambda }^+ \ \left| \begin{array}{lll} \text{ Maximize } &{} \alpha &{} \\ \text{ subject } \text{ to } &{} \omega ^+_{ii} \le \lambda _i &{} (i=1,2,\ldots ,n) \\ &{} \omega ^+_{ij} \le 0 &{} (1 \le i < j \le n) \\ &{} \displaystyle { \left[ \sum _{1 \le k \le l \le n} \omega ^+_{kl} \varPi _+(p_k,p_l) \right] _{ij} \ge \alpha } &{} (1 \le i \le j \le n) \end{array} \right. \end{aligned}$$
(25)
$$\begin{aligned}&\text{(LP) }_{P,\varLambda }^{\pm } \ \left| \begin{array}{lll} \text{ Maximize } &{} \alpha &{} \\ \text{ subject } \text{ to } &{} \omega ^+_{ii} \le \lambda _i &{} (i=1,2,\ldots ,n) \\ &{} \omega ^+_{ij} \le 0, \ \omega ^-_{ij} \le 0 &{} (1 \le i< j \le n) \\ &{} \displaystyle { \left[ \sum _{1 \le k \le l \le n} \omega ^+_{kl} \varPi _+(p_k,p_l) + \sum _{1 \le k < l \le n} \omega ^-_{kl} \varPi _-(p_k,p_l) \right] _{ij} \ge \alpha } &{} (1 \le i \le j \le n) \end{array} \right. \nonumber \\ \end{aligned}$$
(26)
Problem \(\text{(LP) }_{P,\varLambda }^+\) has \(n(n+1)/2+1\) variables and \(n(n+1)\) constraints, and problem \(\text{(LP) }_{P,\varLambda }^{\pm }\) has \(n^2 +1\) variables and \(n(3n+1)/2\) constraints (see Table 1 ). Since \([P\varOmega P^T]_{ij}\) in (10) is given by \(\left[ \sum _{k=1}^n \omega _{kk} \varPi _+(p_k,p_k) \right] _{ij}\), we can prove that both linear optimization problems \(\text{(LP) }_{P,\varLambda }^+\) and \(\text{(LP) }_{P,\varLambda }^{\pm }\) are feasible and bounded by making arguments similar to the one for \(\text{(LP) }_{P,\varLambda }\). Thus, \(\text{(LP) }_{P,\varLambda }^+\) and \(\text{(LP) }_{P,\varLambda }^{\pm }\) have optimal solutions with corresponding optimal values \(\alpha _*^+(P, \varLambda )\) and \(\alpha _*^{\pm }(P, \varLambda )\).
Table 1

Sizes of LPs for identification

Identification

\((P,\varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {G}_n}(A)\)

\((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{+}}(A)\)

\((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}\)

(or \((P,\varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {G}_n}}(A)\))

(or \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{+}}}(A)\))

(or \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{\pm }}}(A)\))

# of variables

\(n+1\)

\(n(n+1)/2 + 1\)

\(n^2+1\)

# of constraints

\(n(n+3)/2\)

\(n(n+1)\)

\(n(3n+1)/2\)

If the optimal value \(\alpha _*^+(P, \varLambda )\) of \(\text{(LP) }_{P,\varLambda }^+\) is nonnegative, then, by rearranging (23), the optimal solution \(\omega ^{+*}_{ij} \ (1 \le i \le j \le n)\) can be made to give the following decomposition:
$$\begin{aligned} A= & {} \left[ \sum _{i=1}^n (\lambda _i-\omega ^{+*}_{ii}) \varPi _+(p_i, p_i) + \sum _{1 \le i < j \le n} (- \omega ^{+*}_{ij}) \varPi _+(p_i, p_j) \right] \\&+ \left[ \sum _{1 \le i \le j \le n} \omega ^{+*}_{ij} \varPi _+(p_i, p_j) \right] \in \mathcal {S}_n^+ + \mathcal {N}_n. \end{aligned}$$
In the same way, if the optimal value \(\alpha _*^{\pm }(P, \varLambda )\) of \(\text{(LP) }_{P,\varLambda }^{\pm }\) is nonnegative, then, by rearranging (24), the optimal solution \(\omega ^{+*}_{ij} \ (1 \le i \le j \le n)\), \(\omega ^{-*}_{ij} \ (1 \le i < j \le n)\) can be made to give the following decomposition:
$$\begin{aligned} A= & {} \left[ \sum _{i=1}^n (\lambda _i-\omega ^{+*}_{ii}) \varPi _+(p_i, p_i) + \sum _{1 \le i< j \le n} (- \omega ^{+*}_{ij}) \varPi _+(p_i, p_j) + \sum _{1 \le i< j \le n} (- \omega ^{-*}_{ij}) \varPi _-(p_i, p_j) \right] \\&+ \left[ \sum _{1 \le i \le j \le n} \omega ^{+*}_{ij} \varPi _+(p_i, p_j) + \sum _{1 \le i < j \le n} \omega ^{-*}_{ij} \varPi _-(p_i, p_j) \right] \ \ \in \mathcal {S}_n^+ + \mathcal {N}_n. \end{aligned}$$
On the basis of the above observations, we can define new subcones of \(\mathcal {S}_n^+ + \mathcal {N}_n\) in a similar manner as (11) and (13).
For a given \(A \in \mathcal {S}_n\), define the following four sets of pairs of matrices
$$\begin{aligned} \begin{array}{rcl} {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{+}}(A) &{} := &{} \{ (P, \varLambda ) \in \mathcal{O}_n \times \mathcal{D}_n \mid P \text{ and } \varLambda \text{ satisfy } (6) \text{ and } \alpha _*^{+}(P,\varLambda ) \ge 0 \} \\ {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}(A) &{} := &{} \{ (P, \varLambda ) \in \mathcal{O}_n \times \mathcal{D}_n \mid P \text{ and } \varLambda \text{ satisfy } (6) \text{ and } \alpha _*^{\pm }(P,\varLambda ) \ge 0 \} \\ {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{+}}}(A) &{} := &{} \{ (P, \varLambda ) \in \mathbb {R}^{n\times n} \times \mathcal{D}_n \mid P \text{ and } \varLambda \text{ satisfy } (6) \text{ and } \alpha _*^{+}(P,\varLambda ) \ge 0 \} \\ {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{\pm }}}(A) &{} := &{} \{ (P, \varLambda ) \in \mathbb {R}^{n\times n} \times \mathcal{D}_n \mid P \text{ and } \varLambda \text{ satisfy } (6) \text{ and } \alpha _*^{\pm }(P,\varLambda ) \ge 0 \} \end{array}\quad \end{aligned}$$
(27)
where \(\alpha _*^{+}(P,\varLambda )\) and \(\alpha _*^{\pm }(P,\varLambda )\) are optimal values of \(\text{(LP) }_{P,\varLambda }^+\) and \(\text{(LP) }_{P,\varLambda }^{\pm }\), respectively. Using the above sets, we define new subcones of \(\mathcal {S}_n^+ + \mathcal {N}_n\) as follows:
$$\begin{aligned} \begin{array}{rcl} \mathcal {F}_n^{+} &{} := &{} \{A \in \mathcal {S}_n \mid {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{+}}(A) \ne \emptyset \}, \\ \mathcal {F}_n^{\pm } &{} := &{} \{A \in \mathcal {S}_n \mid {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}(A) \ne \emptyset \}, \\ \widehat{\mathcal {F}_n^{+}} &{} := &{} \{A \in \mathcal {S}_n \mid {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{+}}}(A) \ne \emptyset \}, \\ \widehat{\mathcal {F}_n^{\pm }} &{} := &{} \{A \in \mathcal {S}_n \mid {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{\pm }}}(A) \ne \emptyset \}. \end{array} \end{aligned}$$
(28)
From the construction of problems \(\text{(LP) }_{P,\varLambda }\), \(\text{(LP) }_{P,\varLambda }^+\) and \(\text{(LP) }_{P,\varLambda }^{\pm }\), and the definitions (27) and (28), we can easily see that
$$\begin{aligned} \mathcal {G}_n \subseteq \mathcal {F}_n^{+} \subseteq \mathcal {F}_n^{\pm }, \ \ \ \ \ \widehat{\mathcal {G}_n} \subseteq \widehat{\mathcal {F}_n^{+}} \subseteq \widehat{\mathcal {F}_n^{\pm }}, \ \ \ \ \ \mathcal {F}_n^{+} \subseteq \widehat{\mathcal {F}_n^{+}}, \ \ \ \ \ \mathcal {F}_n^{\pm } \subseteq \widehat{\mathcal {F}_n^{\pm }} \end{aligned}$$
hold. The corollary below follows from (iv)–(vi) of Theorem 2 and the above inclusions.

Corollary 1

  1. (i)
     
  2. (ii)

    If \(n = 2\), then each of the sets \(\mathcal {F}_n^{+}\), \(\widehat{\mathcal {F}_n^{+}}\), \(\mathcal {F}_n^{\pm }\), and \( \widehat{\mathcal {F}_n^{\pm }}\) coincides with \(\mathcal {S}_n^+ + \mathcal {N}_n\).

     
  3. (iii)

    The convex hull of each of the sets \(\mathcal {F}_n^{+}\), \(\widehat{\mathcal {F}_n^{+}}\), \(\mathcal {F}_n^{\pm }\), and \( \widehat{\mathcal {F}_n^{\pm }}\) is \(\mathcal {S}_n^+ + \mathcal {N}_n\).

     

The following table summarizes the sizes of LPs (10), (25), and (26) that we have to solve in order to identify, respectively, \((P,\varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {G}_n}(A)\) (or \((P,\varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {G}_n}}(A)\)), \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{+}}(A)\) (or \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{+}}}(A)\)), and \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}\) (or \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {F}_n^{\pm }}}(A)\)).

4 Identification of \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\)

In this section, we investigate the effect of using the sets \(\mathcal {G}_n\), \(\mathcal {F}_n^{+}\) and \(\mathcal {F}_n^{\pm }\) for identification of the fact \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\).

We generated random instances of \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) by using the method described in Section 2 of Bomze and Eichfelder (2013). For an \(n \times n\) matrix B with entries independently drawn from a standard normal distribution, we obtained a random positive semidefinite matrix \(S = BB^T\). An \(n \times n\) random nonnegative matrix N was constructed using \(N = C - c_{\min } I_n\) with \(C = F + F^T\) for a random matrix F with entries uniformly distributed in [0, 1] and \(c_{\min }\) being the minimal diagonal entry of C. We set \(A = S + N \in \mathcal {S}_n^+ + \mathcal {N}_n\). The construction was designed so as to maintain the nonnegativity of N while increasing the chance that \(S + N\) would be indefinite and thereby avoid instances that are too easy.

For each instance \(A \in \mathcal {S}^+_n+\mathcal {N}_n\), we used the MATLAB command “\([P, \varLambda ] = \text{ eig } (A) \)” and obtained \((P, \varLambda ) \in \mathcal {O}_n \times \mathcal {D}_n\). We checked whether \((P, \lambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {G}_n}\) (\((P,L) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{+}}\) and \((P,L) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}\)) by solving (\(\text{ LP })_{P,\varLambda }\) in (10) ( (\(\text{ LP })_{P,\varLambda }^+\) in (25) and (\(\text{ LP })_{P,\varLambda }^{\pm }\) in (26)) and if it held, we identified that \(A \in \mathcal {G}_n\) (\(A \in \mathcal {F}_n^{+}\) and \(A \in \mathcal {F}_n^{\pm }\)).

Table 2 shows the number of matrices (denoted by “#A”) that were identified as \(A \in \mathcal {H}_n\) (\(A \in \mathcal {G}_n^{+}\), \(A \in \mathcal {F}_n^{+}\), \(A \in \mathcal {F}_n^{\pm }\) and \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\)) and the average CPU time (denoted by “A.t.(s)”), where 1000 matrices were generated for each n. We used a 3.07GHz Core i7 machine with 12 GB of RAM and Gurobi 6.5 for solving LPs. Note that we performed the last identification \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) as a reference, while we used SeDuMi 1.3 with MATLAB R2015a for solving the semidefinite program (3). The table yields the following observations:
  • All of the matrices were identified as \(A \in \mathcal {S}^+_n+\mathcal {N}_n\) by checking \((P,L) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}\). The result is comparable to the one in Section 2 of Bomze and Eichfelder (2013). The average CPU time for checking \((P,L) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {F}_n^{\pm }}\) is faster than the one for solving the semidefinite program (3) when \(n \ge 20\).

  • For any n, the number of identified matrices increases in the order of the set inclusion relation: \(\mathcal {G}_n \subseteq \mathcal {F}_n^{+} \subseteq \mathcal {F}_n^{\pm }\), while the result for \(\mathcal {H}_n \not \subseteq \mathcal {G}_n\) is better than the one for \(\mathcal {G}_n\) when \(n=10\).

  • For the sets \(\mathcal {H}_n\), \(\mathcal {G}_n\) and \(\mathcal {F}_n^{+}\), the number of identified matrices decreases as the size of n increases.

Table 2

Results of identification of \(A \in \mathcal {S}^+_n+\mathcal {N}_n\): 1000 matrices were generated for each n

n

\(\mathcal {H}_n\)

\(\mathcal {G}_n\)

\(\mathcal {F}_n^{+}\)

\(\mathcal {F}_n^{\pm }\)

\(\mathcal {S}_n^+ + \mathcal {N}_n\)

#A

A.t.(s)

#A

A.t.(s)

#A

A.t.(s)

#A

A.t.(s)

#A

A.t.(s)

10

791

0.001

247

0.005

856

0.008

1000

0.011

1000

0.824

20

16

0.001

20

0.013

719

0.121

1000

0.222

1000

9.282

50

0

0.003

0

2.374

440

22.346

1000

50.092

1000

1285.981

5 LP-based algorithms for testing \(A \in \mathcal {COP}_n\)

In this section, we investigate the effect of using the sets \(\mathcal {F}_n^{+}\), \(\widehat{\mathcal {F}_n^{+}}\), \(\mathcal {F}_n^{\pm }\) and \(\widehat{\mathcal {F}_n^{\pm }}\) for testing whether a given matrix A is copositive by using Sponsel, Bundfuss, and Dür’s algorithm (Sponsel et al. 2012).

5.1 Outline of the algorithms

By defining the standard simplex \(\varDelta ^S\) by \(\varDelta ^S=\{x \in \mathbb {R}^n_+ \mid e^Tx =1 \}\), we can see that a given \(n \times n\) symmetric matrix A is copositive if and only if
$$\begin{aligned} x^TAx \ge 0 \ \text{ for } \text{ all } \ x \in \varDelta ^S \end{aligned}$$
(see Lemma 1 of Bundfuss and Dür 2008). For an arbitrary simplex \(\varDelta \), a family of simplices \(\mathcal {P}=\{\varDelta ^1, \ldots , \varDelta ^m\}\) is called a simplicial partition of \(\varDelta \) if it satisfies
$$\begin{aligned} \varDelta =\bigcup _{i=1}^m \varDelta ^i \ \text{ and } \ \text{ int }(\varDelta ^i)\cap \text{ int }(\varDelta ^j) = \emptyset \ \text{ for } \text{ all } \ i \ne j. \end{aligned}$$
Such a partition can be generated by successively bisecting simplices in the partition. For a given simplex \(\varDelta =\text{ conv }\{v_1, \ldots , v_n\}\), consider the midpoint \(v_{n+1}=\frac{1}{2}(v_i+v_j)\) of the edge \([v_i, v_j]\). Then the subdivision \(\varDelta ^1=\{v_1, \ldots , v_{i-1}, v_{n+1}, v_{i+1}, \ldots , v_n\}\) and \(\varDelta ^2=\{v_1, \ldots , v_{j-1}, v_{n+1}, v_{j+1}, \ldots , v_n\}\) of \(\varDelta \) satisfies the above conditions for simplicial partitions. See Horst (1997) for a detailed description of simplicial partitions.
Denote the set of vertices of partition \(\mathcal {P}\) by
$$\begin{aligned} V(\mathcal {P})=\{v \mid v \text{ is } \text{ a } \text{ vertex } \text{ of } \text{ some } \varDelta \in \mathcal {P} \}. \end{aligned}$$
Each simplex \(\varDelta \) is determined by its vertices and can be represented by a matrix \(V_\varDelta \) whose columns are these vertices. Note that \(V_\varDelta \) is nonsingular and unique up to a permutation of its columns, which does not affect the argument (Sponsel et al. 2012). Define the set of all matrices corresponding to simplices in partition \(\mathcal {P}\) as
$$\begin{aligned} M(\mathcal {P})=\{V_\varDelta :\varDelta \in \mathcal {P} \}. \end{aligned}$$
The “fineness” of a partition \(\mathcal {P}\) is quantified by the maximum diameter of a simplex in \(\mathcal {P}\), denoted by
$$\begin{aligned} \delta (\mathcal {P})=\max _{\varDelta \in \mathcal {P}}\max _{u, v \in \varDelta }||u-v||. \end{aligned}$$
(29)
The above notation was used to show the following necessary and sufficient conditions for copositivity in Sponsel et al. (2012). The first theorem gives a sufficient condition for copositivity.

Theorem 4

[Theorem 2.1 of Sponsel et al. (2012)] If \(A \in \mathcal {S}_n\) satisfies
$$\begin{aligned} V^TAV \in \mathcal {COP}_n \ \text{ for } \text{ all } \ V \in M(\mathcal {P}) \end{aligned}$$
then A is copositive. Hence, for any \(\mathcal {M}_n \subseteq \mathcal {COP}_n\), if \(A \in \mathcal {S}_n\) satisfies
$$\begin{aligned} V^TAV \in \mathcal {M}_n \ \text{ for } \text{ all } \ V \in M(\mathcal {P}), \end{aligned}$$
then A is also copositive.

The above theorem implies that by choosing \(\mathcal {M}_n = \mathcal {N}_n\) (see (2)), A is copositive if \(V^T_\varDelta AV_\varDelta \in \mathcal {N}_n\) holds for any \(\varDelta \in \mathcal {P}\).

Theorem 5

[Theorem 2.2 of Sponsel et al. (2012)] Let \(A \in \mathcal {S}_n\) be strictly copositive, i.e., \(A \in \mathrm{int}\,(\mathcal {COP}_n)\). Then there exists \(\varepsilon > 0\) such that for all partitions \(\mathcal {P}\) of \(\varDelta ^S\) with \(\delta (\mathcal {P}) < \varepsilon \), we have
$$\begin{aligned} V^TAV \in \mathcal {N}_n \ \text{ for } \text{ all } \ V \in M(\mathcal {P}). \end{aligned}$$

The above theorem ensures that if A is strictly copositive (i.e., \(A \in \mathrm{int}\,(\mathcal {COP}_n)\)), the copositivity of A (i.e., \(A \in \mathcal {COP}_n\)) can be detected in finitely many iterations of an algorithm employing a subdivision rule with \(\delta (\mathcal {P}) \rightarrow 0\). A similar result can be obtained for the case \(A \not \in \mathcal {COP}_n\), as follows.

Lemma 2

[Lemma 2.3 of Sponsel et al. (2012)]

The following two statements are equivalent.
  1. 1.

    \(A \notin \mathcal {COP}_n\)

     
  2. 2.

    There is an \(\varepsilon > 0\) such that for any partition \(\mathcal {P}\) with \(\delta (\mathcal {P})<\varepsilon \), there exists a vertex \(v \in V(\mathcal {P})\) such that \(v^TAv<0\).

     
The following algorithm, from Sponsel et al. (2012), is based on the above three results.

As we have already observed, Theorem 5 and Lemma 2 imply the following corollary.

Corollary 2

  1. 1.

    If A is strictly copositive, i.e., \(A \in \mathrm{int}\,(\mathcal {COP}_n)\), then Algorithm 1 terminates finitely, returning “A is copositive.”

     
  2. 2.

    If A is not copositive, i.e., \(A \not \in \mathcal {COP}_n\), then Algorithm 1 terminates finitely, returning “A is not copositive.”

     

In this section, we investigate the effect of using the sets \(\mathcal {H}_n\) from (5), \(\mathcal {G}_n\) from (11), and \(\mathcal {F}_n^{+}\) and \(\mathcal {F}_n^{\pm }\) from (28) as the set \(\mathcal {M}_n\) in the above algorithm.

At Line 7, we can check whether \(V_\varDelta ^TAV_\varDelta \in \mathcal {M}_n\) directly in the case where \(\mathcal {M}_n = \mathcal {H}_n\). In other cases, we diagonalize \(V_\varDelta ^TAV_\varDelta \) as \(V_\varDelta ^TAV_\varDelta = P \varLambda P^T\) and check whether \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {M}_n}(V_\varDelta ^TAV_\varDelta )\) according to definitions (12) or (27). If the associated LP has the nonnegative optimal value, then we identify \(A \in \mathcal {M}_n\).

At Line 8, Algorithm 1 removes the simplex that was determined at Line 7 to be in no further need of exploration by Theorem 4. The accuracy and speed of the determination influence the total computational time and depend on the choice of the set \(\mathcal {M}_n \subseteq \mathcal {COP}_n\).

Here, if we choose \(\mathcal {M}_n = \mathcal {G}_n\) (respectively, \(\mathcal {M}_n = \mathcal {F}_n^{+}\), \(\mathcal {M}_n = \mathcal {F}_n^{\pm }\)), we can improve Algorithm 1 by incorporating the set \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {G}_n}\) (respectively, \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {F}_n^{+}}\), \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {F}_n^{\pm }}\)), as proposed in Tanaka and Yoshise (2015).

The details of the added steps are as follows. Suppose that we have a diagonalization of the form (6).

At Line 8, we need to solve an additional LP but do not need to diagonalize \(V_\varDelta ^TAV_\varDelta \). Let P and \(\varLambda \) be matrices satisfying (6). Then the matrix \(V_{\varDelta }^TP\) can be used to diagonalize \(V_\varDelta ^TAV_\varDelta \), i.e.,
$$\begin{aligned} V_\varDelta ^TAV_\varDelta = V_\varDelta ^T(P\varLambda P^T) V_{\varDelta } = (V_\varDelta ^TP) \varLambda (V_{\varDelta }^TP)^T \end{aligned}$$
while \(V_{\varDelta }^TP \in \mathbb {R}^{n \times n}\) is not necessarily orthogonal. Thus, we can test whether \((V_{\varDelta }^TP, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {M}_n}}\) by solving the corresponding LP according to the definitions (14) or (27). If \((V_{\varDelta }^TP, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {M}_n}}\) holds, then we can identify \(V_\varDelta ^TAV_\varDelta \in \widehat{\mathcal {M}_n}\)

If \((V_{\varDelta }^TP, \varLambda ) \not \in {\mathcal {P}}{\mathcal {L}}_{\widehat{\mathcal {M}_n}}\) at Line 8, we proceed to the original step to identify whether \(V_\varDelta ^TAV_\varDelta \in \mathcal {M}_n\) at Line 12. Similarly to Line 7 of Algorithm 1, we diagonalize \(V_\varDelta ^TAV_\varDelta \) as \(V_\varDelta ^TAV_\varDelta = P\varLambda P^T\) with an orthogonal matrix P and a diagonal matrix \(\varLambda \). Then we check whether \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {M}_n}\) by solving the corresponding LP, and if \((P, \varLambda ) \in {\mathcal {P}}{\mathcal {L}}_{\mathcal {M}_n}\), we can identify \(V_\varDelta ^TAV_\varDelta \in \mathcal {M}_n\).

At Line 18, we don’t need to diagonalize \(V_{\varDelta ^p}^TAV_{\varDelta ^p}\) or solve any more LPs. Let \(\omega ^* \in \mathbb {R}^n\) be an optimal solution of the corresponding LP obtained at Line 8 and let \(\varOmega ^* := \mathrm{Diag}\,(\omega ^*)\). Then the feasibility of \(\omega ^*\) implies the positive semidefiniteness of the matrix \(V_{\varDelta ^p}^TP(\varLambda - \varOmega ^*)P^TV_{\varDelta ^p}\). Thus, if \(V_{\varDelta ^p}^TP\varOmega ^*P^TV_{\varDelta ^p} \in \mathcal {N}_n\), we see that
$$\begin{aligned} V_{\varDelta ^p}^TAV_{\varDelta ^p} = V_{\varDelta ^p}^TP(\varLambda - \varOmega ^*)P^TV_{\varDelta ^p} + V_{\varDelta ^p}^TP\varOmega ^*P^TV_{\varDelta ^p} \in \mathcal {S}_n^+ + \mathcal {N}_n \end{aligned}$$
and that \(V_{\varDelta ^p}^TAV_{\varDelta ^p} \in \widehat{\mathcal {M}_n}\).

5.2 Numerical results

This subsection describes experiments for testing copositivity using \(\mathcal {N}_n\),\(\mathcal {H}_n\), \(\mathcal {G}_n\), \(\mathcal {F}_n^{+}\), \(\widehat{\mathcal {F}_n^{+}}\), \(\mathcal {F}_n^{\pm }\) or \(\widehat{\mathcal {F}_n^{\pm }}\) as the set \(\mathcal {M}_n\) in Algorithms 1 and 2. We implemented the following seven algorithms in MATLAB R2015a on a 3.07 GHz Core i7 machine with 12 GB of RAM, using Gurobi 6.5 for solving LPs:
Algorithm 1.1:

Algorithm 1 with \(\mathcal {M}_n = \mathcal {N}_n\).

Algorithm 1.2:

Algorithm 1 with \(\mathcal {M}_n = \mathcal {H}_n\).

Algorithm 2.1:

Algorithm 2 with \(\mathcal {M}_n = \mathcal {G}_n\) and \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {G}_n}\).

Algorithm 1.3:

Algorithm 1 with \(\mathcal {M}_n = \mathcal {F}_n^{+}\).

Algorithm 2.2:

Algorithm 2 with \(\mathcal {M}_n = \mathcal {F}_n^{+}\) and \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {F}_n^{+}}\).

Algorithm 2.3:

Algorithm 2 with \(\mathcal {M}_n = \mathcal {F}_n^{\pm }\) and \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {F}_n^{\pm }}\).

Algorithm 1.4:

Algorithm 1 with \(\mathcal {M}_n = \mathcal {S}_n^{+}+\mathcal {N}_n\).

As test instances, we used the two kinds of matrices arising from the maximum clique problem (Sect. 5.2.1) and from standard quadratic optimization problems (Sect. 5.2.2).

5.2.1 Results for the matrix arising from the maximum clique problem

In this subsection, we consider the matrix
$$\begin{aligned} B_{\gamma } := \gamma (E -A_G) -E \end{aligned}$$
(30)
where \(E \in \mathcal {S}_n\) is the matrix whose elements are all ones and the matrix \(A_G \in \mathcal {S}_n\) is the adjacency matrix of a given undirected graph G with n nodes. The matrix \(B_\gamma \) comes from the maximum clique problem. The maximum clique problem is to find a clique (complete subgraph) of maximum cardinality in G. It has been shown (in Klerk and Pasechnik 2002) that the maximum cardinality, the so-called clique number \(\omega (G)\), is equal to the optimal value of
$$\begin{aligned} \omega (G) = \min \{ \gamma \in \mathbb {N} \mid B_{\gamma } \in \mathcal {COP}_n \}. \end{aligned}$$
Thus, the clique number can be found by checking the copositivity of \(B_{\gamma }\) for at most \(\gamma =n,n-1, \ldots , 1\).
Figure 3 shows the instances of G that were used in Sponsel et al. (2012). We know the clique numbers of \(G_{8}\) and \(G_{12}\) are \(\omega (G_{8}) = 3\) and \(\omega (G_{12}) = 4\), respectively.
Fig. 3

Graphs \(G_{8}\) with \(\omega (G_{8})=3\) (left) and \(G_{12}\) with \(\omega (G_{12})=4\) (right)

The aim of the implementation is to explore the differences in behavior when using \(\mathcal {H}_n\), \(\mathcal {G}_n\), \(\mathcal {F}_n^{+}\), \(\widehat{\mathcal {F}_n^{+}}\), \(\mathcal {F}_n^{\pm }\) or \(\widehat{\mathcal {F}_n^{\pm }}\) as the set \(\mathcal {M}_n\) rather than to compute the clique number efficiently. Hence, the experiment examined \(B_{\gamma }\) for various values of \(\gamma \) at intervals of 0.1 around the value \(\omega (G)\) (see Tables 3, 4).
Table 3

Results for \(B_{\gamma }\) with \(G_{8}\)

\(\gamma \)

Alg. 1.1 \((\mathcal {N}_n)\)

Alg. 1.2  (\(\mathcal {H}_n\))

Alg. 2.1  (\(\mathcal {G}_n\) , \(\widehat{\mathcal {G}_n}\))

Alg. 1.3  (\(\mathcal {F}_n^{+}\))

Alg. 2.2  (\(\mathcal {F}_n^{+}\) ,  \(\widehat{\mathcal {F}_n^{+}}\))

Alg. 2.3  (\(\mathcal {F}_n^{\pm }\) , \(\widehat{\mathcal {F}_n^{\pm }}\))

Alg. 1.4  (\(\mathcal {S}_n^+\) + \(\mathcal {N}_n\))

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

1.5

1

0.001

1

0.001

1

0.004

1

0.004

1

0.006

1

0.007

1

0.177

1.6

1

0.001

1

0.001

1

0.004

1

0.004

1

0.006

1

0.008

1

0.139

1.7

1

0.001

1

0.001

1

0.004

1

0.004

1

0.006

1

0.007

1

0.180

1.8

1

0.001

1

0.001

1

0.004

1

0.004

1

0.006

1

0.007

1

0.151

1.9

1

0.001

1

0.001

1

0.004

1

0.004

1

0.006

1

0.007

1

0.225

2.0

159

0.018

159

0.021

159

0.454

159

0.368

159

0.690

159

0.964

159

31.782

2.1

159

0.019

159

0.020

159

0.455

159

0.362

159

0.689

159

0.976

159

31.237

2.2

159

0.019

159

0.020

159

0.454

159

0.363

159

0.698

159

0.971

159

31.352

2.3

159

0.019

159

0.020

159

0.452

159

0.364

159

0.689

159

0.964

159

30.115

2.4

159

0.019

159

0.020

159

0.450

159

0.362

159

0.685

159

0.957

159

30.354

2.5

159

0.019

159

0.020

159

0.453

159

0.364

159

0.690

159

0.964

159

29.681

2.6

159

0.019

159

0.020

159

0.453

159

0.361

159

0.684

159

0.953

159

30.060

2.7

2942

0.495

2422

0.346

2687

7.313

2553

5.623

2461

9.772

2307

12.480

1613

293.441

2.8

2942

0.492

2246

0.301

2463

7.197

1951

4.524

1811

7.355

1635

8.731

1251

448.201

2.9

2942

0.501

1606

0.191

2139

6.270

1493

3.469

1393

5.458

1309

6.867

1251

449.572

3.0

3.1

263,548

261.285

3003

0.279

5885

14.603

1827

3.864

1357

4.879

503

2.394

7

3.186

3.2

255,202

243.819

1509

0.132

3129

7.830

911

1.980

377

1.347

201

0.976

3

1.480

3.3

70,814

24.332

469

0.040

2229

5.549

447

0.968

249

0.918

111

0.538

3

1.352

3.4

70,814

23.735

395

0.034

1603

4.112

291

0.625

167

0.650

53

0.254

3

1.401

3.5

70,814

23.821

369

0.031

1

0.003

1

0.003

1

0.004

1

0.004

1

0.322

3.6

70,814

24.304

209

0.017

1

0.002

1

0.003

1

0.004

1

0.004

1

0.362

3.7

70,814

24.302

115

0.009

1

0.002

1

0.003

1

0.004

1

0.004

1

0.371

3.8

70,814

23.744

79

0.007

1

0.002

1

0.003

1

0.004

1

0.004

1

0.359

3.9

70,814

24.101

63

0.005

1

0.002

1

0.003

1

0.003

1

0.005

1

0.322

4.0

70,814

24.242

47

0.004

227

0.593

1

0.003

1

0.003

1

0.005

1

0.360

4.1

4660

0.431

23

0.002

1

0.003

1

0.003

1

0.003

1

0.005

1

0.324

4.2

4660

0.434

17

0.002

1

0.005

1

0.003

1

0.003

1

0.005

1

0.330

4.3

4660

0.432

17

0.002

1

0.005

1

0.003

1

0.003

1

0.005

1

0.324

4.4

4660

0.433

7

0.001

1

0.005

1

0.003

1

0.003

1

0.005

1

0.328

4.5

4660

0.435

7

0.001

1

0.005

1

0.003

1

0.003

1

0.006

1

0.258

Table 4

Results for \(B_{\gamma }\) with \(G_{12}\)

\(\gamma \)

Alg. 1.1  \((\mathcal {N}_n)\)

Alg. 1.2  (\(\mathcal {H}_n\))

Alg. 2.1  (\(\mathcal {G}_n\) , \(\widehat{\mathcal {G}_n}\))

Alg. 1.3  (\(\mathcal {F}_n^{+}\))

Alg. 2.2  (\(\mathcal {F}_n^{+}\) , \(\widehat{\mathcal {F}_n^{+}}\))

Alg. 2.3  (\(\mathcal {F}_n^{\pm }\) , \(\widehat{\mathcal {F}_n^{\pm }}\))

Alg. 1.4  (\(\mathcal {S}_n^+\) + \(\mathcal {N}_n\))

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

2.0

2111

0.483

2111

0.512

2111

8.500

2111

13.782

2111

26.713

2111

49.377

2111

2468.863

2.1

2111

0.483

2111

0.504

2111

8.561

2111

13.682

2111

26.726

2111

49.184

2111

2466.339

2.2

2111

0.482

2111

0.500

2111

8.495

2111

13.585

2111

26.273

2111

48.592

2111

2471.254

2.3

2111

0.483

2111

0.502

2111

8.565

2111

13.520

2111

26.261

2111

48.528

2111

2461.198

2.4

2111

0.489

2111

0.501

2111

8.573

2111

13.315

2111

25.846

2111

47.172

2111

2464.471

2.5

2111

0.485

2111

0.502

2111

8.558

2111

13.344

2111

26.037

2111

47.616

2111

2467.267

2.6

2111

0.481

2111

0.501

2111

8.522

2111

13.304

2111

25.958

2111

47.223

2111

2464.474

2.7

4097

1.136

4097

1.173

4097

16.648

4097

25.753

4097

49.782

4097

90.116

4097

4554.881

2.8

4097

1.136

4097

1.171

4097

16.752

4097

25.655

4097

49.553

4097

89.579

4097

4559.754

2.9

4097

1.134

4097

1.168

4097

16.779

4097

25.643

4097

49.263

4097

89.391

4097

4557.203

3.0

4097

1.134

4097

1.167

4097

16.652

4097

25.550

4097

48.423

4097

87.842

4097

4551.061

3.1

4097

1.136

4097

1.166

4097

16.751

4097

25.484

4097

48.789

4097

87.737

4097

4555.928

3.2

4097

1.138

4097

1.175

4097

16.734

4097

25.422

4097

48.855

4091

87.169

4090

4548.440

3.3

4097

1.131

4095

1.167

4097

16.784

4097

25.274

4091

48.418

4089

86.574

4085

4522.061

3.4

4097

1.131

4087

1.161

4097

16.969

4097

25.179

4091

48.164

4085

86.034

4085

4521.749

3.5

4097

1.139

4085

1.164

4097

16.731

4087

25.122

4087

48.091

4085

85.644

4085

4539.213

3.6

4097

1.137

4085

1.161

4091

16.721

4087

24.920

4087

47.638

4085

84.635

4085

4533.101

3.7

4097

1.140

4085

1.158

4091

16.755

4087

24.834

4085

47.310

4085

84.454

4031

4384.409

3.8

4097

1.133

4084

1.162

4089

17.128

4087

24.831

4085

48.094

4075

85.390

4023

4853.335

3.9

4097

1.137

4080

1.187

4089

17.144

4081

24.719

4079

47.219

4051

84.028

4023

5004.118

4.0

4.1

827,717

18,054.273

4013

5589.341

4.2

899,627

14,932.525

296,637

5093.561

345

528.262

4.3

1,024,493

15,985.310

469,665

6007.219

102,211

1559.054

39

50.717

4.4

1,467,851

16,744.884

592,539

6657.898

147,363

1361.774

36,937

545.801

21

26.4293

4.5

1,125,035

9820.911

354,083

3066.114

66,819

559.987

14,533

213.376

17

20.961

4.6

762,931

5680.756

1,107,483

14,991.047

213,485

1506.465

25,675

206.767

4603

69.503

7

8.768

4.7

610,071

4319.490

793,739

8137.410

125,747

768.224

22,119

180.072

1957

30.490

3

3.809

4.8

569,661

3799.361

473,137

3413.271

69,887

386.279

20,997

176.279

645

10.347

3

4.051

4.9

407,201

1834.912

232,295

1231.091

39,091

207.440

1969

16.716

109

1.889

1

1.221

5.0

305,627

974.829

190,185

859.674

21,283

112.276

1213

10.501

1

0.014

1

1.189

5.1

206,949

415.090

34,641

113.631

12,165

64.742

219

2.000

1

0.013

1

1.150

5.2

141,383

172.541

1

0.004

1

0.008

1

0.008

1

0.012

1

1.120

5.3

110,641

101.475

1

0.003

1

0.008

1

0.007

1

0.012

1

1.040

5.4

90,877

67.681

1

0.003

1

0.008

1

0.008

1

0.012

1

1.078

5.5

44,731

14.292

1

0.003

1

0.007

1

0.007

1

0.011

1

1.100

5.6

26,171

5.910

1

0.004

1

0.007

1

0.007

1

0.012

1

1.000

5.7

15,045

2.775

1

0.004

1

0.008

1

0.008

1

0.012

1

1.057

5.8

10,239

1.705

1

0.003

1

0.007

1

0.007

1

0.012

1

1.063

5.9

6977

1.042

1

0.003

1

0.007

1

0.007

1

0.011

1

1.051

6.0

4717

0.654

1

0.006

1

0.007

1

0.008

1

0.012

1

1.119

As already mentioned, \(\alpha _*(P,\varLambda ) < 0\) (\(\alpha ^+_*(P,\varLambda ) < 0\) and \(\alpha ^{\pm }_*(P,\varLambda ) < 0\)) with a specific P does not necessarily guarantee that \(A \not \in \mathcal {G}_n\) or \(A \not \in \widehat{\mathcal {G}_n}\) (\(A \not \in \mathcal {F}_n^{+}\) or \(A \not \in \widehat{\mathcal {F}_n^{+}}\), \(A \not \in \mathcal {F}_n^{\pm }\) or \(A \not \in \widehat{\mathcal {F}_n^{\pm }}\)). Thus, it not strictly accurate to say that we can use those sets for \(\mathcal {M}_n\), and the algorithms may miss some of the \(\varDelta \)’s that could otherwise have been removed. However, although this may have some effect on speed, it does not affect the termination of the algorithm, as it is guaranteed by the subdivision rule satisfying \(\delta (\mathcal {P}) \rightarrow 0\), where \(\delta (\mathcal {P})\) is defined by (29).
Fig. 4

Graph of Table 4: iterations versus \(\gamma \) of Algorithms 1.2, 2.1, 2.3 and 1.4 for the graph \(G_{12}\)

Tables 3 and 4 show the numerical results for \(G_{8}\) and \(G_{12}\), respectively. Both tables compare the results of the following seven algorithms in terms of the number of iterations (the column “Iter.”) and the total computational time (the column “Time (s)” ):

The symbol “−” means that the algorithm did not terminate within 6 h. The reason for the long computation time may come from the fact that for each graph G, the matrix \(B_{\gamma }\) lies on the boundary of the copositive cone \(\mathcal {COP}_n\) when \(\gamma = \omega (G)\) (\(\omega (G_8)=3\) and \(\omega (G_{12})=4\)). See also Fig. 4, which shows a graph of the results of Algorithms 1.2, 2.1, 2.3, and 1.4 for the graph \(G_{12}\) in Table 4.

We can draw the following implications from the results in Table 4 for the larger graph \(G_{12}\) (similar implications can be drawn from Table 3):
  • At any \(\gamma \ge 5.2\), Algorithms 2.1, 1,3, 2.2, 2.3, and 1.4 terminate in one iteration, and the execution times of Algorithms 2.1, 1.3, 2.2, and 2.3 are much shorter than those of Algorithms 1.1, 1.2, or 1.4.

  • The lower bound of \(\gamma \) for which the algorithm terminates in one iteration and the one for which the algorithm terminates in 6 h decrease in going from Algorithm 1.3 to Algorithm 3.1. The reason may be that, as shown in Corollary 1, the set inclusion relation \(\mathcal {G}_n \subseteq \mathcal {F}_n^{+} \subseteq \mathcal {F}_n^{\pm } \subseteq \mathcal {S}_n^{+}+\mathcal {N}_n\) holds.

  • Table 1 summarizes the sizes of the LPs for identification. The results here imply that the computational times for solving an LP have the following magnitude relationship for any \(n \ge 3\):
    $$\begin{aligned}&\text{ Algorithm } \text{2.1 }< \text{ Algorithm } \text{1.3 }< \text{ Algorithm } \text{2.2 } < \text{ Algorithm } \text{2.3 } . \end{aligned}$$
    On the other hand, the set inclusion relation \(\mathcal {G}_n \subseteq \mathcal {F}_n^{+} \subseteq \mathcal {F}_n^{\pm }\) and the construction of Algorithms 1 and 2 imply that the detection abilities of the algorithms also follow the relationship described above and that the number of iterations has the reverse relationship for any \(\gamma \)s in Table 4:
    $$\begin{aligned}&\text{ Algorithm } \text{2.1 }> \text{ Algorithm } \text{1.3 }> \text{ Algorithm } \text{2.2 } > \text{ Algorithm } \text{2.3 } . \end{aligned}$$
    It seems that the order of the number of iterations has a stronger influence on the total computational time than the order of the computational times for solving an LP.
  • At each \(\gamma \in [4.1, 4.9]\), the number of iterations of Algorithm 2.3 is much larger than one hundred times those of Algorithm 1.4. This means that the total computational time of Algorithm 2.3 is longer than that of Algorithm 1.3 at each \(\gamma \in [4.1, 4.9]\), while Algorithm 1.4 solves a semidefinite program of size \(O(n^2)\) at each iteration.

  • At each \(\gamma < 4\), the algorithms show no significant differences in terms of the number of iterations. The reason may be that they all work to find a \(v \in V(\{\varDelta \})\) such that \(v^T(\gamma (E-A_G)-E)v<0\), while their computational time depends on the choice of simplex refinement strategy.

Fig. 5

Graph of Table 4: time (s) versus \(\gamma \) of Algorithms 1.2, 2.1, 2.3 and 1.4 for the graph \(G_{12}\)

In view of the above observations, we conclude that Algorithm 2.3 with the choices \(\mathcal {M}_n = \mathcal {F}_n^{\pm }\) and \(\widehat{\mathcal {M}_n} = \widehat{\mathcal {F}_n^{\pm }}\) might be a way to check the copositivity of a given matrix A when A is strictly copositive.

The above results are in contrast with those of Bomze and Eichfelder (2013), where the authors show the number of iterations required by their algorithm for testing copositivity of matrices of the form (30). On the contrary to the first observation described above, their algorithm terminates with few iterations when \(\gamma < \omega (G)\), i.e., the corresponding matrix is not copositive, and it requires a huge number of iterations otherwise (Fig. 5).

It should be noted that Table 3 shows an interesting result concerning the non-convexity of the set \(\mathcal {G}_n\), while we know that \(\mathrm{conv}\,(\mathcal {G}_n) = \mathcal {S}_n^+ + \mathcal {N}_n\) (see Theorem 2). Let us look at the result at \(\gamma = 4.0\) of Algorithm 2.1. The multiple iterations at \(\gamma = 4.0\) imply that we could not find \(B_{4.0} \in \mathcal {G}_n\) at the first iteration for a certain orthogonal matrix P satisfying (6). Recall that the matrix \(B_{\gamma }\) is given by (30). It follows from \(E - A_G \in \mathcal {N}_n \subseteq \mathcal {G}_n\) and from the result at \(\gamma = 3.5\) in Table 3 that
$$\begin{aligned} 0.5 (E - A_G) \in \mathcal {G}_n \ \text{ and } \ B_{3,5} = 3.5 (E -A_G) -E \in \mathcal {G}_n. \end{aligned}$$
Thus, the fact that we could not determine whether the matrix
$$\begin{aligned} B_{4.0} = 4.0 (E - A_G) - E = 0.5 (E - A_G) + B_{3.5} \end{aligned}$$
lies in the set \(\mathcal {G}_n\) suggests that the set \(\mathcal {G}_n = \text{ com }(\mathcal {S}_n^+, \mathcal {N}_n)\) is not convex.

5.2.2 Results for the matrix arising from standard quadratic optimization problems

In this subsection, we consider the matrix
$$\begin{aligned} C_{\gamma } := Q - \gamma E \end{aligned}$$
(31)
where \(E \in \mathcal {S}_n\) is the matrix whose elements are all ones and \(Q \in \mathcal {S}_n\) is an arbitrary symmetric matrix, not necessarily positive semidefinite. The matrix \(C_\gamma \) comes from standard quadratic optimization problems of the form,
$$\begin{aligned} \begin{array}{ll} \text{ Minimize } &{} x^TQx \\ \text{ subject } \text{ to } &{} x \in \varDelta ^S := \{ x \in \mathbb {R}^n_+ \mid e^Tx = 1\}. \\ \end{array} \end{aligned}$$
(32)
In Bomze et al. (2000), it is shown that the optimal value of the problem
$$\begin{aligned} p^*(Q) = \max \{ \gamma \in \mathbb {R} \mid C_{\gamma } \in \mathcal {COP}_n \}. \end{aligned}$$
is equal to the optimal value of (32).

The instances of the form (32) were generated using the procedure random_qp in Nowak (1998) with two quartets of parameters \((n, s, k, d) = (10, 5, 5. 0.5)\) and \((n, s, k, d) = (20, 10, 10. 0.5)\), where the parameter n implies the size of Q, i.e., Q is an \(n \times n\) matrix. It has been shown in Nowak (1998) that random_qp generates problems, for which we know the optimal value and a global minimizer a priori for each. We set the optimal value as \(-10\) for each quartet of parameters.

Tables 5 and 6 show the numerical results for \((n, s, k, d) = (10, 5, 5, 0.5)\) and \((n, s, k, d) = (20, 10, 10, 0.5)\). We generated 2 instances for each quartet of parameters and performed the seven algorithms for these instances. Both tables compare the average values of the seven algorithms in terms of the number of iterations (the column “Iter.”) and the total computational time (the column “Time (s)” ): the symbol “−” means that the algorithm did not terminate within 30 minutes. In each table, we made the interval between the values \(\gamma \) smaller as \(\gamma \) got closer to the optimal value, to observe the behavior around the optimal value more precisely.
Table 5

Results for \(C_{\gamma }\) with \(n=10\)

\(\gamma \)

Alg. 1.1 \((\mathcal {N}_n)\)

Alg. 1.2 (\(\mathcal {H}_n\))

Alg. 2.1 (\(\mathcal {G}_n\) , \(\widehat{\mathcal {G}_n}\))

Alg. 1.3 (\(\mathcal {F}_n^{+}\))

Alg. 2.2 (\(\mathcal {F}_n^{+}\) , \(\widehat{\mathcal {F}_n^{+}}\))

Alg. 2.3 (\(\mathcal {F}_n^{\pm }\) , \(\widehat{\mathcal {F}_n^{\pm }}\))

Alg. 1.4 (\(\mathcal {S}_n^+\) + \(\mathcal {N}_n\))

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

\(-\) 5.00000

1

0.001

1

0.001

1

0.009

1

0.009

1

0.016

1

0.025

1

0.501

\(-\) 7.50000

1

0.001

1

0.001

1

0.010

1

0.009

1

0.016

1

0.025

1

0.491

\(-\) 8.75000

1

0.001

1

0.001

1

0.010

1

0.008

1

0.016

1

0.025

1

0.481

\(-\) 9.37500

2

.001

2

0.001

2

0.020

2

0.018

2

0.032

2

0.048

2

0.946

\(-\) 9.68750

3

0.001

3

0.001

3

0.021

3

0.021

3

0.041

3

0.066

3

1.621

\(-\) 9.84375

799

0.267

799

0.259

799

4.993

799

4.815

799

9.253

799

15.329

799

441.790

\(-\) 10.00000

\(-\) 10.15625

84,828

81.527

1

0.001

1

0.029

1

0.007

1

0.011

1

0.012

1

0.424

\(-\) 10.31250

39,790

17.030

1

0.001

1

0.004

1

0.008

1

0.011

1

0.012

1

0.362

\(-\) 10.62500

5546

1.210

1

0.001

1

0.005

1

0.008

1

0.011

1

0.012

1

0.380

\(-\) 11.25000

8

0.007

1

0.001

1

0.005

1

0.007

1

0.011

1

0.012

1

0.363

\(-\) 12.50000

2

0.001

1

0.001

1

0.005

1

0.007

1

0.011

1

0.012

1

0.367

\(-\) 15.00000

2

0.001

1

0.001

1

0.005

1

0.008

1

0.011

1

0.012

1

0.384

Table 6

Results for \(C_{\gamma }\) with \(n=20\)

\(\gamma \)

Alg. 1.1 \((\mathcal {N}_n)\)

Alg. 1.2 (\(\mathcal {H}_n\))

Alg. 2.1 (\(\mathcal {G}_n\) , \(\widehat{\mathcal {G}_n}\))

Alg. 1.3 (\(\mathcal {F}_n^{+}\))

Alg. 2.2 (\(\mathcal {F}_n^{+}\) , \(\widehat{\mathcal {F}_n^{+}}\))

Alg. 2.3 (\(\mathcal {F}_n^{\pm }\) , \(\widehat{\mathcal {F}_n^{\pm }}\))

Alg. 1.4 (\(\mathcal {S}_n^+\) + \(\mathcal {N}_n\))

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

Iter.

Time (s)

\(-\) 5.00000

1

0.002

1

0.002

1

0.042

1

0.095

1

0.181

1

0.182

1

13.812

\(-\) 7.50000

1

0.002

1

0.002

1

0.045

1

0.093

1

0.183

1

0.180

1

14.266

\(-\) 8.75000

2

0.005

2

0.005

2

0.046

2

0.191

2

0.368

2

0.378

2

25.224

\(-\) 9.37500

128

0.091

128

0.096

128

2.490

128

11.755

128

22.660

128

22.682

128

1598.031

\(-\) 9.68750

\(-\) 9.84375

\(-\) 10.00000

\(-\) 10.15625

1

0.080

1

0.080

1

0.076

1

13.912

\(-\) 10.31250

1

0.010

1

0.072

1

0.072

1

0.073

1

14.015

\(-\) 10.62500

2

0.001

1

0.009

1

0.072

1

0.069

1

0.073

1

14.135

\(-\) 11.25000

2

0.001

1

0.009

1

0.067

1

0.067

1

0.071

1

12.126

\(-\) 12.50000

3187

1.370

2

0.001

1

0.010

1

0.068

1

0.066

1

0.067

1

11.128

\(-\) 15.00000

2

0.001

2

0.001

1

0.010

1

0.069

1

0.080

1

0.068

1

9.580

From the results in Tables 5 and 6, we can draw implications that are very similar to those for the maximum clique problem, listed (we hence, omitted discussing them here). A major difference from the implications for the maximum clique problem is that Algorithm 1.2 using the set \(\mathcal {H}_n\) is efficient for solving a small (\(n=10\)) standard quadratic problem, while it cannot solve the problem within 30 minutes when \(n=20\) and \(\gamma \ge -10.3125\).

6 Concluding remarks

In this paper, we investigated the properties of several tractable subcones of \(\mathcal {S}_n^+ + \mathcal {N}_n\) and summarized the results (as Figs. 12). We also devised new subcones of \(\mathcal {S}_n^+ + \mathcal {N}_n\) by introducing the semidefinite basis (SD basis) defined as in Definitions 1 and 2. We conducted numerical experiments using those subcones for identification of given matrices \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) and for testing the copositivity of matrices arising from the maximum clique problem and from standard quadratic optimization problems. We have to solve LPs with \(O(n^2)\) variables and \(O(n^2)\) constraints in order to detect whether a given matrix belongs to those cones, and the computational cost is substantial. However, the numerical results shown in Tables 2, 3, 4 and 6 show that the new subcones are promising not only for identification of \(A \in \mathcal {S}_n^+ + \mathcal {N}_n\) but also for testing copositivity.

Recently, Ahmadi et al. (2015) developed algorithms for inner approximating the cone of positive semidefinite matrices, wherein they focused on the set \(\mathcal {D}_n \subseteq \mathcal {S}_n^+\) of \(n \times n\) diagonal dominant matrices. Let \(U_{n,k}\) be the set of vectors in \(\mathbb {R}^n\) that have at most k nonzero components, each equal to \(\pm 1\), and define
$$\begin{aligned} \mathcal {U}_{n,k} := \{ uu^T \mid u \in U_{n,k} \}. \end{aligned}$$
Then, as the authors indicate, the following theorem has already been proven.

Theorem 6

[Theorem 3.1 of Ahmadi et al. (2015), Barker and Carlson Barker and Carlson (1975)]
$$\begin{aligned} \mathcal {D}_n = \text{ cone }(\mathcal {U}_{n,k}) := \left\{ \sum _{i=1}^{|\mathcal {U}_{n,k}|} \alpha _i U_i \mid U_i \in \mathcal {U}_{n,k}, \ \ \alpha _i \ge 0 \ (i=1, \ldots , |\mathcal {U}_{n,k}|) \right\} \end{aligned}$$
From the above theorem, we can see that for the SDP bases \(\mathcal {B}_+(p_1, p_2, \ldots , p_n)\) in (20), \(\mathcal {B}_-(p_1, p_2, \ldots , p_n)\) in (22) and n-dimensional unit vectors \(e_1, e_2, \ldots , e_n\), the following set inclusion relation holds:
$$\begin{aligned} \mathcal {B}_+(e_1, e_2, \ldots , e_n) \cup \mathcal {B}_-(e_1, e_2, \ldots , e_n) \subseteq \mathcal {D}_n = \text{ cone }(\mathcal {U}_{n,k}). \end{aligned}$$
These sets should be investigated in the future.

Notes

Acknowledgements

The authors would like to sincerely thank the anonymous reviewers for their thoughtful and valuable comments which have significantly improved the paper. Among others, one of the reviewers pointed out that Proposition 2 is Lemma 6.2 of Dickinson (2011), suggested to revise the title of the paper, and the definitions of the sets \(\mathcal {G}_n\), \(\mathcal {F}_n^{+}\), etc., to be more accurate, and gave the second example in (16).

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan
  2. 2.Faculty of Engineering, Information and SystemsUniversity of TsukubaTsukubaJapan

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