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Connections Between Spectral Properties of Asymptotic Mappings and Solutions to Wireless Network Problems

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2019年第10期67卷 2747-2760页

作者： Cavalcante, Renato L. G. Liao, Qi Stanczak, Slawomir Tech Univ Berlin Fraunhofer Heinrich Hertz Inst D-10623 Berlin Germany Nokia Bell Labs D-70435 Stuttgart Germany

In this study, we establish connections between asymptotic functions and properties of solutions to important problems in wireless networks. We start by introducing a class of self-mappings (called asymptotic mappings) constructed with asymptotic functions, and we show that spectral properties of these mappings explain the behavior of solutions to some max-min utility optimization problems. For example, in a common family of max-min utility power control problems, we prove that the optimal utility as a function of the power available to transmitters is approximately linear in the low power regime. However, as we move away from this regime, there exists a transition point, easily computed from the spectral radius of an asymptotic mapping, from which gains in utility become increasingly marginal. From these results we derive analogous properties of the transmit energy efficiency. In this study, we also generalize and unify existing approaches for feasibility analysis in wireless networks. Feasibility problems often reduce to determining the existence of the fixed point of a standard interference mapping, and we show that the spectral radius of an asymptotic mapping provides a necessary and sufficient condition for the existence of such a fixed point. We further present a result that determines whether the fixed point satisfies a constraint given in terms of a monotone norm.

关键词： Utility optimization fixed point algorithms interference functions asymptotic mappings nonconvex optimization power control interference management

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WEAKLY STANDARD INTERFERENCE MAPPINGS: EXISTENCE OF fixed pointS AND APPLICATIONS TO POWER CONTROL IN WIRELESS NETWORKS 44

WEAKLY STANDARD INTERFERENCE MAPPINGS: EXISTENCE OF FIXED PO...

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44th IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Cavalcante, R. L. G. Stanczak, S. Fraunhofer Heinrich Hertz Inst Berlin Germany Tech Univ Berlin Berlin Germany

ISBN: (纸本)9781479981311

We propose novel approaches to identify the existence of fixed points of the so-called weakly standard interference mappings, which include the well-known standard and general interference mappings as particular cases. The approaches are based on the concept of spectral radius of asymptotic mappings, a mathematical tool recently introduced to study the behavior of wireless networks. We show that, for arbitrary weakly standard interference mappings, knowledge of the spectral radius of an associated asymptotic mapping gives a sufficient condition to determine the existence of fixed points or their absence in the positive orthant. If the mapping has a fixed point, we further prove that the set of fixed points has a minimal element that can be easily computed with a simple fixed point algorithm. The theory developed here is applied to the problem of power control for load planning in LTE networks. Unlike previous approaches in the literature, the proposed solution takes into account the limited number of modulation and coding schemes of practical transceivers.

关键词： Interference mappings power control asymptotic mappings fixed point algorithms

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IMPROVED PRECISION OF fixed-point algorithms BY MEANS OF COMMON FACTORS

IMPROVED PRECISION OF FIXED-POINT ALGORITHMS BY MEANS OF COM...

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15th IEEE International Conference on Image Processing (ICIP 2008)

作者： Reznik, Yuriy A. Hinds, Arianne T. Mitchell, Joan L. Stanford Univ Informat Syst Lab 353 Serra Mall Stanford CA 94305 USA IBM Info Print Solut Co Ricoh Boulder CO 80301 USA

ISBN: (纸本)9781424417650

We describe a general technique for improving the precision of fixed-point implementations of signal processing algorithms (such as filters, transforms, etc.) relying on the use of "common factors". Such factors are applied to groups of real constants in the algorithms (e.g. filter coefficients), turning them into quantities that can be more accurately approximated by dyadic rational numbers. We show that the problem of optimal design of such approximations is related to the classic Diophantine approximation problem, and explain how it can be solved and used for improving practical designs.

关键词： Signal processing fixed point algorithms Diophantine approximations

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A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid

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MATHEMATICAL PROGRAMMING 2009年第1期118卷 13-35页

作者： Beck, Amir Teboulle, Marc Technion Israel Inst Technol Dept Ind Engn IL-32000 Haifa Israel Tel Aviv Univ Sch Math Sci IL-69978 Tel Aviv Israel

We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem's class we study. We prove that under a certain mild assumption on the problem's data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance epsilon grows as O(root ln epsilon(-1)).

关键词： Ratio of quadratic minimization Nonconvex quadratic minimization Semidefinite programming Strong duality Regularized total least squares fixed point algorithms Convergence analysis

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On the Convergence of ICA algorithms With Symmetric Orthogonalization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2009年第6期57卷 2209-2221页

作者： Erdogan, Alper T. Koc Univ Dept Elect Engn TR-34450 Istanbul Turkey

Independent component analysis (ICA) problem is often posed as the maximization/minimization of an objective/cost function under a unitary constraint, which presumes the prewhitening of the observed mixtures. The parallel adaptive algorithms corresponding to this optimization setting, where all the separators are jointly trained, are typically implemented by a gradient-based update of the separation matrix followed by the so-called symmetrical orthogonalization procedure to impose the unitary constraint. This article addresses the convergence analysis of such algorithms, which has been considered as a difficult task due to the complication caused by the minimum-(Frobenius or induced 2-norm) distance mapping step. We first provide a general characterization of the stationary points corresponding to these algorithms. Furthermore, we show that fixed point algorithms employing symmetrical orthogonalization are monotonically convergent for convex objective functions. We later generalize this convergence result for nonconvex objective functions. At the last part of the article, we concentrate on the kurtosis objective function as a special case. We provide a new set of critical points based on Householder reflection and we also provide the analysis for the minima/maxima/saddle-point classification of these critical points.

关键词： Blind source separation convergence fixed point algorithms independent component analysis (ICA) symmetric orthogonalization

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On the convergence of ICA algorithms with symmetric orthogonalization

On the convergence of ICA algorithms with symmetric orthogon...

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33rd IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Erdogan, Alper T. Koc Univ EE Dept Istanbul Turkey

ISBN: (纸本)9781424414833

We study the convergence behavior of Independent Component Analysis (ICA) algorithms that are based on the contrast function maximization and that employ symmetric orthogonalization method to guarantee the orthogonality property of the search matrix. In particular, the characterization of the critical points of the corresponding optimization problem and the stationary points of the conventional gradient ascent and fixed point algorithms are obtained. As an interesting and a useful feature of the symmetrical orthogonalization method, we show that the use of symmetric orthogonalization enables the monotonic convergence for the fixed point ICA algorithms that are based on the convex contrast functions.

关键词： Independent Component Analysis Blind Source Separation symmetric orthogonalization fixed point algorithms convergence

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Shadow prices for LLR and ALBA

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IEEE-ACM TRANSACTIONS ON NETWORKING 1996年第5期4卷 796-807页

作者： Vargas, C Hegde, MV NaraghiPour, M Min, PS LOUISIANA STATE UNIV DEPT ELECT & COMP ENGNBATON ROUGELA 70803 WASHINGTON UNIV DEPT ELECT ENGNST LOUISMO 63130

Shadow prices are calculated for least loaded routing (LLR) and aggregated least busy alternative (ALBA) routing in circuit-switched networks for the blocking probability obtained from fixed point algorithms, Numerical results are presented for the calculation of these shadow prices in small networks. As an application of these shadow prices, we also formulate a constrained optimization problem to calculate the sum capacity of LLR and ALBA for a given network, Comparison of the sum capacities indicate that the optimization using shadow prices results in a significant improvement, This provides evidence that matching capacity distribution to traffic is important even when adaptive routing schemes such as LLR and ALBA are used in the network, We also calculate upper bounds on the sum capacity which serve to indicate how well the optimized LLR and ALBA perform, The numerical results also confirm that with a small number of states the capacity of ALBA approaches that of LLR.

关键词： Circuit Switching Telecommunication Network Routing Telecommunication Traffic Probability Optimisation LLR ALBA Least Loaded Routing Aggregated Least Busy Alternative Routing Circuit Switched Networks Blocking Probability fixed point algorithms Shadow Prices Constrained Optimization Problem Sum Capacity Capacity Distribution Traffic Adaptive Routing Routing Telecommunication Traffic Constraint Optimization Switching Circuits Pricing Network Topology Probability Upper Bound Intelligent Networks Switches

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A CONSTRUCTIVE PROOF OF A PERMUTATION-BASED GENERALIZATION OF SPERNERS LEMMA

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MATHEMATICAL PROGRAMMING 1989年第1期44卷 113-120页

作者： BAPAT, RB 1.Indian Statistical Institute 7 S.J.S. Sansanwal Marg 110016 New Delhi India

In a recent paper, Gale has given an interesting generalization of the KKM lemma in combinatorial topology. We present a similar generalization of Sperner's well-known lemma and give a constructive proof. The argument uses the familiar idea of following simplicial paths in a triangulation. To demonstrate that the algorithm must work, orientation considerations are necessary. Gale's generalized KKM lemma is derived from the main result. A permutation-based generalization of Brouwer's fixed point theorem is also given.

关键词： Sperner's lemma simplicial pivoting fixed point algorithms

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A UNIFIED APPROACH TO THE IMPLEMENTATION OF SEVERAL RESTART fixed-point algorithms AND A NEW VARIABLE DIMENSION ALGORITHM

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MATHEMATICAL PROGRAMMING 1984年第3期28卷 288-328页

作者： KOJIMA, M YAMAMOTO, Y UNIV TSUKUBA INST SOCIOECON PLANNINGSAKURAIBARAKI 305JAPAN

We present a unified description of a class of restart fixed point algorithms including Merrill's method and several variable dimension algorithms for their implementation on computers. Based on this description we show that some pivot-saving techniques originally developed for the homotopy methods can be applied to the class. We also propose a new variable dimension algorithm having 3n-1 rays along which we can move toward a solution. Some numerical comparisons of the simplicial restart algorithms, Merrill's method, the 2n-method, the octahedral method and the new one, support that the latter two methods are more efficient than the others.

关键词： fixed point algorithms Systems of Nonlinear Equations Variable Dimension algorithms Piecewise Smooth Mappings Piecewise Linear Mappings Complementary Pivoting

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A PL HOMOTOPY CONTINUATION METHOD WITH THE USE OF AN ODD MAP FOR THE ARTIFICIAL LEVEL

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MATHEMATICAL PROGRAMMING 1985年第2期31卷 235-244页

作者： KOJIMA, M OISHI, S SUMI, Y HORIUCHI, K WASEDA UNIV TOKYO 160JAPAN SHIBAURA INST TECHNOL TOKYOJAPAN

This note presents a new piecewise linear homotopy continuation method for solving a system of nonlinear equations. The important feature of the method is the use of an odd map for the artificial level of the homotopy. Some sufficient conditions for the global convergence of the method are given. They are different from the known conditions for the global convergence of the existing homotopy continuation methods. Specifically, they cover all the systems of nondegenerate linear equations.

关键词： Continuation Method fixed point algorithms Systems of Equations Odd Maps Piecewise Linear Approximation

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