scholarly journals Linear mappings preserving the copositive cone

2021 ◽  
pp. 1
Author(s):  
Yaroslav Shitov
Keyword(s):  
Copositive Cone

scholarly journals The extreme rays of the $$6\times 6$$ copositive cone

2020 ◽  
Author(s):  
Andrey Afonin ◽  
Roland Hildebrand ◽  
Peter J. C. Dickinson
Keyword(s):  
Copositive Cone ◽  
Extreme Rays

scholarly journals On the algebraic structure of the copositive cone

2020 ◽  
Vol 14 (8) ◽  
pp. 2007-2019
Author(s):  
Roland Hildebrand
Keyword(s):  
Algebraic Structure ◽  
Copositive Cone

scholarly journals Irreducible elements of the copositive cone

2013 ◽  
Vol 439 (6) ◽  
pp. 1605-1626 ◽  
Author(s):  
Peter J.C. Dickinson ◽  
Mirjam Dür ◽  
Luuk Gijben ◽  
Roland Hildebrand
Keyword(s):  
Irreducible Elements ◽  
Copositive Cone

scholarly journals On the accuracy of uniform polyhedral approximations of the copositive cone

2012 ◽  
Vol 27 (1) ◽  
pp. 155-173 ◽  
Author(s):  
E. Alper Yıldırım
Keyword(s):  
Copositive Cone ◽  
Polyhedral Approximations

Scaling relationship between the copositive cone and Parrilo’s first level approximation

2012 ◽  
Vol 7 (8) ◽  
pp. 1669-1679 ◽  
Author(s):  
Peter J. C. Dickinson ◽  
Mirjam Dür ◽  
Luuk Gijben ◽  
Roland Hildebrand
Keyword(s):  
Scaling Relationship ◽  
Copositive Cone

scholarly journals An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

2021 ◽  
Author(s):  
Riley Badenbroek ◽  
Etienne de Klerk
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Cutting Plane ◽  
Mixed Integer ◽  
Cutting Plane Method ◽  
Analytic Center ◽  
Completely Positive ◽  
Plane Method ◽  
Copositive Cone

We propose an analytic center cutting plane method to determine whether a matrix is completely positive and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman et al. [Berman A, Dur M, Shaked-Monderer N (2015) Open problems in the theory of completely positive and copositive matrices. Electronic J. Linear Algebra 29(1):46–58]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia et al. [Xia W, Vera JC, Zuluaga LF (2020) Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1):40–56]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like [Formula: see text] for d × d matrices. The method is implemented in Julia and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl . Summary of Contribution: Completely positive matrices play an important role in operations research. They allow many NP-hard problems to be formulated as optimization problems over a proper cone, which enables them to benefit from the duality theory of convex programming. We propose an analytic center cutting plane method to determine whether a matrix is completely positive by solving an optimization problem over the copositive cone. In fact, we can use our method to solve any copositive optimization problem, provided we know the radius of a ball containing an optimal solution. We emphasize numerical performance and stability in developing this method. A software implementation in Julia is provided.

scholarly journals The extreme rays of the 5×5 copositive cone

2012 ◽  
Vol 437 (7) ◽  
pp. 1538-1547 ◽  
Author(s):  
Roland Hildebrand
Keyword(s):  
Copositive Cone ◽  
Extreme Rays

scholarly journals Structural Properties of Faces of the Cone of Copositive Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2698
Author(s):  
Olga Kostyukova ◽  
Tatiana Tchemisova
Keyword(s):  
Structural Properties ◽  
Geometric Structure ◽  
Incomplete Data ◽  
Facial Structure ◽  
Copositive Matrices ◽  
Copositive Cone ◽  
Explicit Representations ◽  
The Given

In this paper, we study the properties of faces and exposed faces of the cone of copositive matrices (copositive cone), paying special attention to issues related to their geometric structure. Based on the concepts of zero and minimal zero vectors, we obtain several explicit representations of faces of the copositive cone and compare them. Given a face of the cone of copositive matrices, we describe the subspace generated by that face and the minimal exposed face containing it. Summarizing the results obtained in the paper, we systematically show what information can be extracted about the given copositive face in the case of incomplete data. Several examples for illustrating the main findings of the paper and also for justifying the usefulness of the developed approach to the study of the facial structure of the copositive cone are discussed.

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