Open problems in the theory of completely positive and copositive matrices

Berman, Abraham; Dür, Mirjam; Shaked-Monderer, Naomi

doi:10.13001/1081-3810.2943

Cited by 66 publications

(76 citation statements)

References 63 publications

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“…for all j according to (7) and the sequence {x t } is bounded according to (i), we deduce further that {η tj } is bounded. By passing to a further subsequence if necessary, we may assume without loss of generality that η tj → η * for some η * .…”

Section: ⊓ ⊔mentioning

confidence: 61%

“…where the last inequality follows from (7) with M = 0, (35), and the facts that x t − x * ≤ ǫ/2 and that F (x * ) < F (x t ) ≤ F (x N 1 ) < F (x * ) + ǫ (since t ≥ N 1 ). Dividing both sides of the above inequality by c 2 , taking square root, using the relation √ ab ≤ a+b 2 for any nonnegative numbers a and b and invoking the definition of C 1 , we obtain further that…”

Section: A Proof Of Theoremmentioning

confidence: 97%

“…We would like to point out that the completely positive rank of G is generally hard to compute (see [7]) and we will specify the r used in our experiments later. In [16], the authors considered two algorithms for solving (27): 1. the classical alternating projection method, which can be inefficient because Proj P is in general difficult to compute;…”

Section: ⊓ ⊔mentioning

confidence: 99%

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A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection

et al. 2020

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The split feasibility problem is to find an element in the intersection of a closed set C and the linear preimage of another closed set D, assuming the projections onto C and D are easy to compute. This class of problems arises naturally in many contemporary applications such as compressed sensing. While the sets C and D are typically assumed to be convex in the literature, in this paper, we allow both sets to be possibly nonconvex. We observe that, in this setting, the split feasibility problem can be formulated as an optimization problem with a difference-of-convex objective so that standard majorization-minimization type algorithms can be applied. Here we focus on the nonmonotone proximal gradient algorithm with majorization studied in [21, Appendix A]. We show that, when this algorithm is applied to a split feasibility problem, the sequence generated clusters at a stationary point of the problem under mild assumptions. We also study local convergence property of the sequence under suitable assumptions on the closed sets involved. Finally, we perform numerical experiments to illustrate

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Section: ⊓ ⊔mentioning

confidence: 61%

Section: A Proof Of Theoremmentioning

confidence: 97%

Section: ⊓ ⊔mentioning

confidence: 99%

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A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection

et al. 2020

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“…, m) is impossible. Very recently, Zhang [29] showed that the CPP cone with dimension not less than 5 is not facially exposed, i.e., some of its faces are non-exposed (see also [6,13] for geometric properties of the CPP cone). Thus, our framework using COP(co(K ∩ J), Q 0 ) is more general than the work using (2) in [1,2,3,4,19].…”

Section: Relations To Existing Workmentioning

confidence: 99%

A Geometrical Analysis on Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

Kim¹,

Kojima

Toh

2020

SIAM J. Optim.

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We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective function in a linear space V, and the constraint set is represented geometrically as the intersection of a nonconvex cone K ⊂ V, a face J of the convex hull of K and a parallel translation L of a supporting hyperplane of the nonconvex cone K. We show that under a moderate assumption, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of J and the hyperplane L. The replacement procedure is applied to derive the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems. Key words.Completely positive reformulation of quadratic and polynomial optimization problems, conic optimization problems, hierarchies of copositivity, faces of the completely positive cone.AMS Classification. 90C20, 90C25, 90C26.widely studied subclass of POPs as they include many important NP-hard combinatorial problems such as binary QOPs, maximum stable set problems, graph partitioning problems and quadratic assignment problems. To numerically solve QOPs, a common approach is through solving their convex conic relaxations such as semidefinite programming relaxations [23,21] and doubly nonnegative (DNN) relaxations [15,19,26,28]. As those relaxations provide lower bounds of different qualities, the tightness of the lower bounds has been a very critical issue in assessing the strength of the relaxations. The completely positive programming (CPP) reformulation of QOPs, which provides their exact optimal values, has been extensively studied in theory. More specifically, QOPs over the standard simplex [9, 10], maximum stable set problems [12], graph partitioning problems [24], and quadratic assignment problems [25] are equivalently reformulated as CPPs. Burer's CPP reformulations [11] of a class of linearly constrained QOPs in nonnegative and binary variables provided a more general framework to study the specific problems mentioned above. See also the papers [1,2,8,14,22] for further developments.Despite a great deal of studies on the CPP relaxation, its geometrical aspects have not been well understood. The main purpose of this paper is to present and analyze essential features of the CPP reformulation of QOPs and its extension to POPs by investigating their geometry. With the geometrical analysis, many existing equivalent reformulations of QOPs and POPs can be considered in a unified manner and deriving effective numerical methods for computing tight bounds can be facilitated. In particular, the class of QOPs that can be equivalently reformulated as CPPs in our framework includes Burer's class of linearly constrained QOPs in nonnegative and...

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“…We use bold letters like v, w ∈ C n to denote vectors, non-bold uppercase letters like A, B ∈ M n (C) to denote matrices, and non-bold lowercase letters like c, d ∈ C to denote scalars. Subscripts on non-bold letters indicate particular entries of a vector or matrix (e.g., v 1 , v 2 , v 3 denote the first 3 coordinates of the vector v), while subscripts of bold letters denote particular vectors (e.g., v 1 , v 2 , v 3 are three vectors). Double subscripts are sometimes used to denote specific entries of vectors that are themselves denoted by subscripts (e.g., v 3,7 refers to the 7-th entry of the vector v 3 ).…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

Johnston

MacLean

2019

ELA

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We introduce a generalization of the set of completely positive matrices that we call "pairwise completely positive" (PCP) matrices. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise nonnegative. We explore basic properties of these matrix pairs and develop several testable necessary and sufficient conditions that help determine whether or not a pair is PCP. We then establish a connection with quantum entanglement by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of our results, we show that a wide family of states that have absolutely positive partial transpose are in fact separable.

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Open problems in the theory of completely positive and copositive matrices

Abstract: Abstract. We describe the main open problems which are currently of interest in the theory of copositive and completely positive matrices. We give motivation as to why these questions are relevant and provide a brief description of the state of the art in each open problem.

Cited by 66 publications

References 63 publications

A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection

A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection

A Geometrical Analysis on Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

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