Geometry of the copositive and completely positive cones

Dickinson, Peter J. C.

doi:10.1016/j.jmaa.2011.03.005

Cited by 51 publications

(30 citation statements)

References 14 publications

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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%

“…. , m) and m. Conversely, we can construct any face J of coK by (12), (13) and (14) as we shall present next. Since all Q pj ∈ V (j = 1, .…”

Section: The Hierarchy Of Copositivity Conditionmentioning

confidence: 99%

“…We also see that if 0 ≤ k < p ≤ m and P ∈ V is copositive on J k , then it is copositive on J p since (J k ) * ⊃ (J p ) * . This implies that replacing (13) by (15) is not restrictive at all. Furthermore, if J p−1 is a face of coK, then J p−1 = co(K ∩ J p−1 ) by (i) of Lemma 3.4.…”

Section: The Hierarchy Of Copositivity Conditionmentioning

confidence: 99%

“…Furthermore, if J p−1 is a face of coK, then J p−1 = co(K ∩ J p−1 ) by (i) of Lemma 3.4. Hence, "copositive on J p−1 " can be replaced by "copositive on K ∩ J p−1 " in (13) and (15). Lemma 4.2.…”

Section: The Hierarchy Of Copositivity Conditionmentioning

confidence: 99%

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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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Section: Relations To Existing Workmentioning

confidence: 99%

“…. , m) and m. Conversely, we can construct any face J of coK by (12), (13) and (14) as we shall present next. Since all Q pj ∈ V (j = 1, .…”

Section: The Hierarchy Of Copositivity Conditionmentioning

confidence: 99%

Section: The Hierarchy Of Copositivity Conditionmentioning

confidence: 99%

Section: The Hierarchy Of Copositivity Conditionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kim¹,

Kojima

Toh

2020

SIAM J. Optim.

View full text Add to dashboard Cite

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“…A cone is facially exposed if all of its faces are exposed. Although every extreme ray of CP d is exposed [59], it remains unknown whether CP d is facially exposed. In the case of COP d , the extreme rays corresponding to |ii ii| are not exposed [59], implying that PSD d + N d (the set of DEWs for PPTDS states) is not facially exposed.…”

Section: B Exposednessmentioning

confidence: 99%

Separability of diagonal symmetric states: a quadratic conic optimization problem

Tura

Aloy

Quesada

et al. 2018

Quantum

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We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in C d ⊗ C d (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for d ≤ 4. Furthermore, for d ≥ 5, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on N -partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

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Exploiting partial correlations in distributionally robust optimization

2019

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In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set {(x, xx ) : x ∈ X } where X is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in Project Evaluation and Review Technique (PERT) networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled.

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Geometry of the copositive and completely positive cones

Cited by 51 publications

References 14 publications

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

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

Separability of diagonal symmetric states: a quadratic conic optimization problem

Exploiting partial correlations in distributionally robust optimization

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