358 research outputs found
Nonnegative Matrix Inequalities and their Application to Nonconvex Power Control Optimization
Maximizing the sum rates in a multiuser Gaussian channel by power control is a nonconvex NP-hard problem that finds engineering application in code division multiple access (CDMA) wireless communication network. In this paper, we extend and apply several fundamental nonnegative matrix inequalities initiated by Friedland and Karlin in a 1975 paper to solve this nonconvex power control optimization problem. Leveraging tools such as the Perron–Frobenius theorem in nonnegative matrix theory, we (1) show that this problem in the power domain can be reformulated as an equivalent convex maximization problem over a closed unbounded convex set in the logarithmic signal-to-interference-noise ratio domain, (2) propose two relaxation techniques that utilize the reformulation problem structure and convexification by Lagrange dual relaxation to compute progressively tight bounds, and (3) propose a global optimization algorithm with ϵ-suboptimality to compute the optimal power control allocation. A byproduct of our analysis is the application of Friedland–Karlin inequalities to inverse problems in nonnegative matrix theory
Generalized Perron--Frobenius Theorem for Nonsquare Matrices
The celebrated Perron--Frobenius (PF) theorem is stated for irreducible
nonnegative square matrices, and provides a simple characterization of their
eigenvectors and eigenvalues. The importance of this theorem stems from the
fact that eigenvalue problems on such matrices arise in many fields of science
and engineering, including dynamical systems theory, economics, statistics and
optimization. However, many real-life scenarios give rise to nonsquare
matrices. A natural question is whether the PF Theorem (along with its
applications) can be generalized to a nonsquare setting. Our paper provides a
generalization of the PF Theorem to nonsquare matrices. The extension can be
interpreted as representing client-server systems with additional degrees of
freedom, where each client may choose between multiple servers that can
cooperate in serving it (while potentially interfering with other clients).
This formulation is motivated by applications to power control in wireless
networks, economics and others, all of which extend known examples for the use
of the original PF Theorem.
We show that the option of cooperation between servers does not improve the
situation, in the sense that in the optimal solution no cooperation is needed,
and only one server needs to serve each client. Hence, the additional power of
having several potential servers per client translates into \emph{choosing} the
best single server and not into \emph{sharing} the load between the servers in
some way, as one might have expected.
The two main contributions of the paper are (i) a generalized PF Theorem that
characterizes the optimal solution for a non-convex nonsquare problem, and (ii)
an algorithm for finding the optimal solution in polynomial time
Convergence of Tomlin's HOTS algorithm
The HOTS algorithm uses the hyperlink structure of the web to compute a
vector of scores with which one can rank web pages. The HOTS vector is the
vector of the exponentials of the dual variables of an optimal flow problem
(the "temperature" of each page). The flow represents an optimal distribution
of web surfers on the web graph in the sense of entropy maximization.
In this paper, we prove the convergence of Tomlin's HOTS algorithm. We first
study a simplified version of the algorithm, which is a fixed point scaling
algorithm designed to solve the matrix balancing problem for nonnegative
irreducible matrices. The proof of convergence is general (nonlinear
Perron-Frobenius theory) and applies to a family of deformations of HOTS. Then,
we address the effective HOTS algorithm, designed by Tomlin for the ranking of
web pages. The model is a network entropy maximization problem generalizing
matrix balancing. We show that, under mild assumptions, the HOTS algorithm
converges with a linear convergence rate. The proof relies on a uniqueness
property of the fixed point and on the existence of a Lyapunov function.
We also show that the coordinate descent algorithm can be used to find the
ideal and effective HOTS vectors and we compare HOTS and coordinate descent on
fragments of the web graph. Our numerical experiments suggest that the
convergence rate of the HOTS algorithm may deteriorate when the size of the
input increases. We thus give a normalized version of HOTS with an
experimentally better convergence rate.Comment: 21 page
The Collatz-Wielandt quotient for pairs of nonnegative operators
In this paper we consider two versions of the Collatz-Wielandt quotient for a
pair of nonnegative operators A,B that map a given pointed generating cone in
the first space into a given pointed generating cone in the second space. If
the two spaces and two cones are identical, and B is the identity operator then
one version of this quotient is the spectral radius of A. In some applications,
as commodity pricing, power control in wireless networks and quantum
information theory, one needs to deal with the Collatz-Wielandt quotient for
two nonnegative operators. In this paper we treat the two important cases: a
pair of rectangular nonnegative matrices and a pair completely positive
operators. We give a characterization of minimal optimal solutions and
polynomially computable bounds on the Collatz-Wielandt quotient.Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-794
Z-matrix equations in max algebra, nonnegative linear algebra and other semirings
We study the max-algebraic analogue of equations involving Z-matrices and
M-matrices, with an outlook to a more general algebraic setting. We show that
these equations can be solved using the Frobenius trace down method in a way
similar to that in non-negative linear algebra, characterizing the solvability
in terms of supports and access relations. We give a description of the
solution set as combination of the least solution and the eigenspace of the
matrix, and provide a general algebraic setting in which this result holds.Comment: 21 pages, 1 figur
Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms
We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton\u27s iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence
Equivalence of finite dimensional input-output models of solute transport and diffusion in geosciences
We show that for a large class of finite dimensional input-output positive
systems that represent networks of transport and diffusion of solute in
geological media, there exist equivalent multi-rate mass transfer and multiple
interacting continua representations, which are quite popular in geo-sciences.
Moreover, we provide explicit methods to construct these equivalent
representations. The proofs show that controllability property is playing a
crucial role for obtaining equivalence. These results contribute to our
fundamental understanding on the effect of fine-scale geological structures on
the transfer and dispersion of solute, and, eventually, on their interaction
with soil microbes and minerals
On Projection-Based Model Reduction of Biochemical Networks-- Part I: The Deterministic Case
This paper addresses the problem of model reduction for dynamical system
models that describe biochemical reaction networks. Inherent in such models are
properties such as stability, positivity and network structure. Ideally these
properties should be preserved by model reduction procedures, although
traditional projection based approaches struggle to do this. We propose a
projection based model reduction algorithm which uses generalised block
diagonal Gramians to preserve structure and positivity. Two algorithms are
presented, one provides more accurate reduced order models, the second provides
easier to simulate reduced order models. The results are illustrated through
numerical examples.Comment: Submitted to 53rd IEEE CD
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