812 research outputs found
An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix
We propose an analytic center cutting plane method to determine if 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, D\"ur, and Shaked-Monderer [Electronic Journal of Linear Algebra,
2015]. 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, Vera, and Zuluaga [INFORMS Journal on Computing, 2018]. 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 for matrices. The method is
implemented in Julia, and available at
https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl.Comment: 16 pages, 1 figur
Stochastic Data Clustering
In 1961 Herbert Simon and Albert Ando published the theory behind the
long-term behavior of a dynamical system that can be described by a nearly
uncoupled matrix. Over the past fifty years this theory has been used in a
variety of contexts, including queueing theory, brain organization, and
ecology. In all these applications, the structure of the system is known and
the point of interest is the various stages the system passes through on its
way to some long-term equilibrium.
This paper looks at this problem from the other direction. That is, we
develop a technique for using the evolution of the system to tell us about its
initial structure, and we use this technique to develop a new algorithm for
data clustering.Comment: 23 page
Simulated annealing with hit-and-run for convex optimization: rigorous complexity analysis and practical perspectives for copositive programming
We give a rigorous complexity analysis of the simulated annealing algorithm
by Kalai and Vempala [Math of OR 31.2 (2006): 253-266] using the type of
temperature update suggested by Abernethy and Hazan [arXiv 1507.02528v2, 2015].
The algorithm only assumes a membership oracle of the feasible set, and we
prove that it returns a solution in polynomial time which is near-optimal with
high probability. Moreover, we propose a number of modifications to improve the
practical performance of this method, and present some numerical results for
test problems from copositive programming
An analytic center cutting plane method to determine complete positivity of a matrix
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 1. 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):40-561 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 O(d(2)) for d x 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
A new graph perspective on max-min fairness in Gaussian parallel channels
In this work we are concerned with the problem of achieving max-min fairness
in Gaussian parallel channels with respect to a general performance function,
including channel capacity or decoding reliability as special cases. As our
central results, we characterize the laws which determine the value of the
achievable max-min fair performance as a function of channel sharing policy and
power allocation (to channels and users). In particular, we show that the
max-min fair performance behaves as a specialized version of the Lovasz
function, or Delsarte bound, of a certain graph induced by channel sharing
combinatorics. We also prove that, in addition to such graph, merely a certain
2-norm distance dependent on the allowable power allocations and used
performance functions, is sufficient for the characterization of max-min fair
performance up to some candidate interval. Our results show also a specific
role played by odd cycles in the graph induced by the channel sharing policy
and we present an interesting relation between max-min fairness in parallel
channels and optimal throughput in an associated interference channel.Comment: 41 pages, 8 figures. submitted to IEEE Transactions on Information
Theory on August the 6th, 200
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