Completely Positive Matrices

Berman, Abraham; Shaked-Monderer, Naomi

doi:10.1142/5273

Cited by 196 publications

(153 citation statements)

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“…The convex hull C = conv xx ⊤ : x ∈ R n + , results in a convex matrix cone called the cone of completely positive matrices since [71]; for a text see [7]. Note that a similar construction dropping nonnegativity constraints leads to P = conv xx ⊤ : x ∈ R n , the cone of positive-semidefinite matrices, the basic set in Semidefinite Optimization (SDP), wherefrom above lifting idea was borrowed.…”

Section: Motivation Notation and Basic Ideasmentioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

134

117

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Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.

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Section: Motivation Notation and Basic Ideasmentioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

134

117

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“…Since then, numerous papers on both copositivity and complete positivity have emerged in the linear algebra literature, see [6] or [36] for surveys. Using these cones in optimization has been studied only in the last decade.…”

Section: Historical Remarksmentioning

confidence: 99%

“…Proofs of all these statements can be found in [6]. The interior of the completely positive cone has first been characterized in [27].…”

Section: Topological Propertiesmentioning

confidence: 99%

“…Most of them use linear algebraic arguments or rely on properties of the graph associated to the matrix, and it seems unclear how they can be used for algorithmic methods to solve optimization problems over C * . For a comprehensible survey of these conditions, we refer to [6]. We just mention two sufficient conditions: a sufficient condition shown in [39] is that A is nonnegative and diagonally dominant.…”

Section: Complete Positivitymentioning

confidence: 99%

“…It can be shown (see e.g. [6]) that C * is the cone of so-called completely positive matrices C * = conv{xx T : x ∈ R n + }.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Copositive Programming – a Survey

Dür

2010

Recent Advances in Optimization and Its Applications in Engineering

178

179

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Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.

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Similarity network aggregation for the analysis of glacier ecosystems

Ambrosini,

Baccini,

Barabesi

2024

Environmetrics

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The synthesis of information deriving from complex networks is a topic receiving increasing relevance in ecology and environmental sciences. In particular, the aggregation of multilayer networks, that is, network structures formed by multiple interacting networks (the layers), constitutes a fast‐growing field. In several environmental applications, the layers of a multilayer network are modeled as a collection of similarity matrices describing how similar pairs of biological entities are, based on different types of features (e.g., biological traits). The present paper first discusses two main techniques for combining the multi‐layered information into a single network (the so‐called monoplex), that is, similarity network fusion and similarity matrix average (SMA). Then, the effectiveness of the two methods is tested on a real‐world dataset of the relative abundance of microbial species in the ecosystems of nine glaciers (four glaciers in the Alps and five in the Andes). A preliminary clustering analysis on the monoplexes obtained with different methods shows the emergence of a tightly connected community formed by species that are typical of cryoconite holes worldwide. Moreover, the weights assigned to different layers by the SMA algorithm suggest that two large South American glaciers (Exploradores and Perito Moreno) are structurally different from the smaller glaciers in both Europe and South America. Overall, these results highlight the importance of integration methods in the discovery of the underlying organizational structure of biological entities in multilayer ecological networks.

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Completely Positive Matrices

Cited by 196 publications

References 0 publications

Copositive optimization – Recent developments and applications

Copositive optimization – Recent developments and applications

Copositive Programming – a Survey

Similarity network aggregation for the analysis of glacier ecosystems

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