A matrix A is called completely positive if it can be decomposed as A = BB^T with an entrywise nonnegative matrix B. The set of all such matrices is a convex cone. We provide a characterization of the interior of this cone as well as of its dual

Dür, Mirjam

Still, Georg

Electronic Journal of Linear Algebra

English

Still, Georg J.

University of Twente Research Information

ELAINTERIOR POINTS OF THE COMPLETELY POSITIVE CONE∗MIRJAM DÜR† AND GEORG STILL‡Abstract. A matrix A is called completely positive if it can be decomposed as A = BBT withan entrywise nonnegative matrix B. The set of all such matrices is a convex cone which plays a rolein certain optimization problems. A characterization of the interior of this cone is provided.Key words. Completely positive matrices, Copositive matrices, Cones of matrices.AMS subject classifications. 15A23, 15A48.1. Introduction. A symmetric matrix A is called completely positive if it al-lows a factorization A = BBT with an entrywise nonnegative matrix B. This classof matrices has received quite an amount of interest in the linear algebra literatureduring the last decades. An excellent survey of this literature is the book [2]. How-ever, completely positive matrices have recently also attracted some interest in themathematical programming community.Since the 1980s, so called semidefinite relaxations have been proposed as a strongmethod to obtain good bounds for many combinatorial optimization problems. Asemidefinite program is an optimization problem where a linear function of a matrixvariable is to be minimized subject to linear constraints and an additional semidefi-niteness constraint, i.e., one wants to optimize over the cone P of positive semidefinitematrices. Efficient algorithms called interior point methods have been developed forthis type of problem. For an introduction to semidefinite programming, see [7].Starting with [3], it has then been observed that some combinatorial problemslike the maximum clique problem can equivalently be reformulated as optimizationproblems over the cone CP of completely positive matrices. Burer [4] showed the verygeneral result that every quadratic problem with linear and binary constraints canbe rewritten as such a problem. More precisely, he showed that a quadratic binary∗Received by the editors October 06, 2007. Accepted for publication January 20, 2008. HandlingEditor: Raphael Loewy.†Technische Universität Darmstadt, Department of Mathematics, Schloßgartenstr. 7, 64289Darmstadt, Germany (duer@mathematik.tu-darmstadt.de).‡University of Twente, Department of Applied Mathematics, P.O.Box 217, 7500 AE Enschede,The Netherlands (g.still@math.utwente.nl).48Electronic Journal of Linear Algebra  ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 17, pp. 48-53, February 2008http://math.technion.ac.il/iic/elaELAInterior Points of the Completely Positive Cone 49problem of the formmin xT Qx + 2cT xs.t. aTi x = bi (i = 1, . . . , m)x ≥ 0xj ∈ {0, 1} (j ∈ B)(with Q not necessarily positive semidefinite) can equivalently be written as the fol-lowing linear problem over the cone of completely positive matrices:min 〈Q, X〉+ 2cT xs.t. aTi x = bi (i = 1, . . . , m)〈aiaTi , X〉 = b2i (i = 1, . . . , m)xj = Xjj (j ∈ B)(1 xx X)∈ CP.This is a remarkable result, since it transforms a nonconvex quadratic integer problemequivalently into a linear problem over a convex cone, i.e., a convex optimizationproblem which has no nonglobal local optima. The difficulty, of course, is now in thecone constraint, whence it is essential to get a better understanding of the cone.The dual problem of a completely positive program is an optimization problemover the cone of copositive matrices. Obviously, both problem classes are NP-hardsince they are equivalent to integer programming.Interior point algorithms have proved to be very efficient for semidefinite prob-lems. Because of the nonpolynomial complexity of completely positive programs (incontrast to polynomial complexity of interior point methods for semidefinite pro-grams) it will not be possible to extend these methods directly to the completelypositive cone. But one might still try to design algorithms which use interior pointsfor a completely positive program. However, nothing seems to be known about thestructure of the interior of CP. This is what we investigate in this note.We use the following notation:S = {A ∈ Rn×n : A = AT }, the cone of symmetric matrices,N = {A ∈ S : A ≥ 0}, the cone of (entrywise) nonnegative matrices,P = {A ∈ S : A  0}, the cone of positive semidefinite matrices,CP = {B = ∑mi=1 aiaTi : ai ≥ 0}, the cone of completely positive matrices,COP = {A ∈ S : xT Ax ≥ 0 ∀x ≥ 0}, the cone of copositive matrices.Electronic Journal of Linear Algebra  ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 17, pp. 48-53, February 2008http://math.technion.ac.il/iic/elaELA50 Mirjam Dür and Georg StillAn equivalent definition is CP = {B = AAT : A ∈ Rn×m, A ≥ 0}. Clearly, thefactorization of a completely positive matrix is not unique.Obviously, we have the following relations:CP ⊆ P ∩N and COP ⊇ P +N .Interestingly, for n×n-matrices of order n ≤ 4, we have equality in the above relations,whereas for n ≥ 5, both inclusions are strict, cf. [6].The inner product in S is defined as 〈A, B〉 := trace(AB). For a given coneK ⊆ S, the dual cone K∗ is defined asK∗ := {A ∈ S : 〈A, B〉 ≥ 0 for all B ∈ K}.It can be shown that K = (K∗)∗ if and only if K is a closed convex cone (see forexample [2, Theorem 1.36]). All matrix cones defined above are closed convex cones.We haveS∗ = {0}, N ∗ = N , P∗ = P , CP∗ = COP , COP∗ = CP.For a proof of the last two relations see for instance [2, Theorem 2.3].It is easy to see that int(N ) = {A ∈ S : A > 0} and int(P) = {A ∈ S : A  0} ={A = BBT : B ∈ Rn×n nonsingular}. For the dual of a closed convex cone K, it hasbeen shown in [1, Chapter 1, Section 2] thatint(K∗) = {A ∈ S : 〈A, B〉 > 0 for all B ∈ K \ {0}}. (1.1)From this relation, it is not difficult to derive a characterization of the interior of COP :we getint(COP) = {A ∈ S : xT Ax > 0 for all x ≥ 0, x = 0},which says that the interior of COP consists of the so called strictly copositive matrices.This is a well known result, but as far as we are aware, no analogous result isknown for the cone CP. The next section provides a characterization of its interior.2. Characterization of the Interior of CP. It follows from the definitionof CP that the inclusionCP ⊆ P ∩N (2.1)always holds. From this, we directly concludeint(CP) ⊆ int(P) ∩ int(N ) = {A ∈ S : A  0, A > 0}, (2.2)Electronic Journal of Linear Algebra  ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 17, pp. 48-53, February 2008http://math.technion.ac.il/iic/elaELAInterior Points of the Completely Positive Cone 51which immediately implies the following:Lemma 2.1. If A ∈ int(CP), then A > 0 and rankA = n.Moreover, since for n ≤ 4 equality holds in (2.1), we must haveint(CP) = {A ∈ S : A  0, A > 0} for n ≤ 4.On the other hand, the fact that for n ≥ 5 the inclusion (2.1) is strict implies that alsothe inclusion (2.2) is strict for n ≥ 5. Indeed, if we choose matrices A ∈ (P ∩N ) \ CPand B ∈ int(P ∩N ), then by convexity,Xλ := A + λ(B − A) ∈ int(P ∩N ) for all 0 < λ ≤ 1 .However, since CP is closed, the relation A /∈ CP implies Xλ /∈ CP for λ > 0 smallenough, so that Xλ ∈ int(P ∩N ) \ CP. We also provide a concrete counterexample.Example 2.2. TakeA =1 0 0 0 1 1 0 0 1 21 1 0 0 0 2 1 0 0 10 1 1 0 0 1 2 1 0 00 0 1 1 0 0 1 2 1 00 0 0 1 1 0 0 1 2 1⇒ AAT =8 5 1 1 55 8 5 1 11 5 8 5 11 1 5 8 55 1 1 5 8.Clearly, AAT > 0 and rankAAT = 5, so AAT ∈ int(P ∩ N ). Nevertheless, thereexists a copositive matrix H such that 〈AAT , H〉 = 0, which, by (1.1), proves thatAAT /∈ int(CP). This matrix is the Horn matrixH =1 −1 1 1 −1−1 1 −1 1 11 −1 1 −1 11 1 −1 1 −1−1 1 1 −1 1which was introduced by Horn to illustrate that there exist copositive matrices whichare not decomposable as the sum of a positive semidefinite and a nonnegative matrix,cf. [5]. To see that H is copositive, writexT Hx = (x1 − x2 + x3 + x4 − x5)2 + 4x2x4 + 4x3(x5 − x4)= (x1 − x2 + x3 − x4 + x5)2 + 4x2x5 + 4x1(x4 − x5).The first expression shows that xT Hx ≥ 0 for nonnegative x with x5 ≥ x4, whereasthe second expression shows xT Hx ≥ 0 for nonnegative x with x5 < x4.Electronic Journal of Linear Algebra  ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 17, pp. 48-53, February 2008http://math.technion.ac.il/iic/elaELA52 Mirjam Dür and Georg StillIn view of the preceding discussions, to find a characterization of int(CP) forarbitrary matrix dimensions we have to look for subsets of the cone{AAT : AAT > 0, A ≥ 0, rankAAT = n}.The next theorem gives such a characterization. We use the notation [A1|A2] todescribe the matrix whose columns are the columns of A1 augmented with the columnsof A2.Theorem 2.3. We have:int(CP) = {AAT : A = [A1|A2] with A1 > 0 nonsingular, A2 ≥ 0}.Proof. Denote M := {AAT : A = [A1|A2] with A1 > 0 nonsingular, A2 ≥ 0} forabbreviation.[M ⊆ int(CP)]: Let B ∈ M. We show that B1 = A1AT1 ∈ int(CP). Then, sinceCP is a cone, also B = A1AT1 + A2AT2 ∈ int(CP).So we only have to show that the statement holds for B = AAT with nonsingular0 < A ∈ Rn×n. To do so, we choose S ∈ S arbitrarily and prove that for any ε smallenough there exists C ∈ Rn×n, C ≥ 0, such thatAAT + εS = CCT . (2.3)The relation (2.3) is equivalent toI + εA−1SA−T = (A−1C)(A−1C)T . (2.4)We put M := A−1SA−T and note that for small ε the matrix I + εM is positivedefinite. It is well-known that by a (symmetric) Gauss elimination algorithm (e.g.the Cholesky decomposition) any positive definite matrix E can be decomposed asE = QQT with nonsingular Q.This transformation Q = Q(E) depends continuously on E. Therefore, since I hasthe obvious decomposition I = QQT with Q(I) = I, also I+εM has a decompositionI + εM = QQT , Q = Q(I + εM) = I + V (ε),with some matrix V (ε) which tends to the zero matrix as ε → 0. Comparing with(2.4), we can put A−1C = Q and we finally obtain a representation (2.3) withC = AQ = A(I + V (ε)) = A + AV (ε) > 0Electronic Journal of Linear Algebra  ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 17, pp. 48-53, February 2008http://math.technion.ac.il/iic/elaELAInterior Points of the Completely Positive Cone 53for all small ε, which proves M ⊆ int(CP).[int(CP) ⊆ M]: Let B be an arbitrary matrix in int(CP), and choose some matrixB = F ∈ M ⊂ CP. Since CP is a convex cone, we can construct a matrix X ∈ CPsuch that B is a strict convex combination of F and X . Indeed, since B ∈ int(CP),there exists α > 1 such that X := F +α(B−F ) ∈ CP. The last equation is equivalenttoB = (1− 1α )F + 1αX = F̃ + X̃.Clearly, F̃ = ÃÃT with Ã = [Ã1|Ã2] with Ã1 > 0 nonsingular, Ã2 ≥ 0, and X̃ = Y Y Tfor some Y ≥ 0 since X̃ ∈ CP. Consequently,B = [Ã1|Ã2|Y ][Ã1|Ã2|Y ]Tis a factorization of B with the required properties, whence B ∈ M.Acknowledgment. We wish to thank the referee for careful reading and helpfulcomments.REFERENCES[1] A. Berman. Cones, Matrices and Mathematical Programming. Lecture Notes in Economics andMathematical Systems 79, Springer Verlag 1973.[2] A. Berman and N. Shaked-Monderer. Completely Positive Matrices. World Scientific, 2003.[3] I.M. Bomze, M. Dür, E. de Klerk, C. Roos, A.J. Quist, and T. Terlaky. On Copositive Program-ming and Standard Quadratic Optimization Problems, Journal of Global Optimization,18:301–320, 2000.[4] S. Burer. On the Copositive Representation of Binary and Continuous Nonconvex QuadraticPrograms, Preprint, University of Iowa, 2006, available online athttp://www.optimization-online.org/DB_HTML/2006/10/1501.html[5] M. Hall Jr. and M. Newman. Copositive and Completely Positive Quadratic Forms, Proceedingsof the Cambridge Philosophical Society, 59:329–33, 1963.[6] J.E. Maxfield and H. Minc. On the Matrix Equation X′X = A, Proceedings of the EdinburghMathematical Society, 13:125–129, 1962/1963.[7] H. Wolkowicz (Ed.). Handbook of Semidefinite Programming: Theory, Algorithms, and Appli-cations. Kluwer Academic Publishers, 2000.Electronic Journal of Linear Algebra  ISSN 1081-3810 A publication of the International Linear Algebra SocietyVolume 17, pp. 48-53, February 2008http://math.technion.ac.il/iic/ela

Interior points of the completely positive cone.

A matrix A is called completely positive if it can be decomposed as A = BBT with an entrywise nonnegative matrix B. The set of all such matrices is a convex cone which plays a role in certain optimization problems. A characterization of the interior of this cone is provided

Mirjam Dür

Georg Still

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INTERIOR POINTS OF THE COMPLETELY POSITIVE CONE

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Interior points of the completely positive cone

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A matrix A is called completely positive if it can be decomposed as A = BBT with&nbsp;an entrywise nonnegative matrix B. The set of all such matrices is a convex cone which plays a role&nbsp;in certain optimization problems. A characterization of the interior of this cone is provided

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Interior points of the completely positive cone.

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