Nonlinear Perron-Frobenius Theory Research Articles

A unified approach to nonlinear Perron-Frobenius theory

Let f:R>0n→R>0n be an order-preserving and homogeneous function. We show that the set of eigenvectors of f in R>0n is nonempty and bounded in Hilbert's projective metric if and only if f satisfies a condition involving upper and lower Collatz-Wielandt numbers of readily computed auxiliary functions. This condition generalizes a test for the existence of eigenvectors using hypergraphs that was proved by Akian, Gaubert, and Hochart. We include several examples to show how the new condition can be combined with the hypergraph test to give a systematic approach to determine when homogeneous and order-preserving functions have eigenvectors in R>0n. We also observe that if the entries of f are real analytic functions on R>0n, then the set of eigenvectors of f in R>0n is nonempty and bounded in Hilbert's projective metric if and only if the eigenvector is unique, up to scaling.

Sum Rate and Max-Min Rate for Cellular-Enabled UAV Swarm Networks

This paper investigates the fundamental rate performances in the highly promising cellular-enabled unmanned aerial vehicle (UAV) swarm networks, which can provide ubiquitous wireless connectivity for supporting various Internet of things (IoT) applications. We first provide the formulations for the sum rate maximization and max-min rate, which are two nonlinear optimization problems subject to the constraints of UAV transmit power and antenna parameters at base station (BS). For the sum rate maximization problem, we propose an iterative algorithm to solve it utilizing the Karush–Kuhn–Tucker (KKT) condition. For the max-min rate problem, we transform it to an equivalent conditional eigenvalue problem based on the nonlinear Perron-Frobenius theory, and thus design an iterative algorithm to obtain the solution of such problem. Finally, numerical results are presented to indicate the effect of some key parameters on the rate performances in such networks.

Nonexpansive maps with surjective displacement

We investigate necessary and sufficient conditions for a nonexpansive map $f$ on a Banach space $X$ to have surjective displacement, that is, for $f - \mathrm{id}$ to map onto $X$. In particular, we give a computable necessary and sufficient condition when $X$ is a finite dimensional space with a polyhedral norm. We give a similar computable necessary and sufficient condition for a fixed point of a polyhedral norm nonexpansive map to be unique. We also consider applications to nonlinear Perron-Frobenius theory and suggest some additional computable sufficient conditions for surjective displacement and uniqueness of fixed points.

A nonlinear Perron–Frobenius approach for stability and consensus of discrete-time multi-agent systems

In this paper we propose a novel method to study stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) where agents evolve with nonlinear dynamics, possibly different for each agent (heterogeneous local interaction rules). In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by positive, sub-homogeneous and type-K order-preserving nonlinear maps. This paper generalizes results that apply to linear MASs to the nonlinear case by exploiting nonlinear Perron–Frobenius theory. We provide sufficient conditions on the structure of the nonlinear local interaction rules to guarantee stability of a MAS and an additional condition on the topology of the network ensuring the achievement of consensus as a particular case. Two examples are provided to corroborate the theoretical analysis. In the first one we consider a susceptible–infected–susceptible (SIS) model while in the second we consider a novel protocol to solve the max-consensus problem.

A framework for second-order eigenvector centralities and clustering coefficients.

We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron-Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.

Fair Optimal Resource Allocation in Cognitive Radio Networks With Co-Channel Interference Mitigation

In this paper, we study the utility fairness resource allocation in a multi-user orthogonal-based cognitive radio network with cochannel interference (CCI) mitigation. In our proposed system model, we introduce the correct reception probability (CRP) model as a network utility metric. Furthermore, useful bounds on CRP are derived to analyze the performance of proposed allocation schemes. The optimal resource allocation is formulated as a worst-case user CRP maximum problem with both average CCI and average power budget constraints. However, this problem is non-convex and generally challenging to solve. Therefore, we solve this problem by successively performing subchannel allocation and power allocation. Firstly, a $k$ -means clustering inspired subchannel allocation strategy is proposed to divide secondary users (SUs) into multiple groups by minimizing the average mutual-signal-to-interference-ratio degree between any two SUs. The concept of reference user is employed to guarantee the quality of service of the primary user. In each subchannel, we formulate a max-min utility optimal power allocation problem. The non-linear Perron Frobenius theory is applied to solve this power allocation problem. Simulation results show that the proposed resource allocation scheme is fair and has relatively fast convergence.

Max-Min Fairness Rate Control in Wireless Networks: Optimality and Algorithms by Perron-Frobenius Theory

Rate adaptation and power control are two key resource allocation mechanisms in multiuser wireless networks. In the presence of interference, how do we jointly optimize end-to-end source rates and link powers to achieve weighted max-min rate fairness for all sources in the network? This optimization problem is hard to solve as physical layer link rate functions are nonlinear, nonconvex, and coupled in the transmit powers. We show that the weighted max-min rate fairness problem can, in fact, be decoupled into separate fairness problems for flow rate and power control. For a large class of physical layer link rate functions, we characterize the optimal solution analytically by a nonlinear Perron-Frobenius theory through solving a conditional eigenvalue problem that captures the interaction of multiuser interference. We propose an iterative algorithm to compute the optimal flow rate that converges geometrically fast without any parameter configuration. Numerical results demonstrate that our iterative algorithm is computationally fast for the Shannon capacity, CDMA, and piecewise link rate functions.

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

We consider the problem of finding an efficient and fair ex-ante rule for division of an uncertain monetary outcome among a finite number of von Neumann–Morgenstern agents. Efficiency is understood here, as usual, in the sense of Pareto efficiency subject to the feasibility constraint. Fairness is defined as financial fairness with respect to a predetermined pricing functional. We show that efficient and financially fair allocation rules are in one-to-one correspondence with positive eigenvectors of a nonlinear homogeneous and monotone mapping associated to the risk sharing problem. We establish relevant properties of this mapping. On the basis of this, we obtain a proof of existence and uniqueness of solutions via nonlinear Perron–Frobenius theory, as well as a proof of global convergence of the natural iterative algorithm. We argue that this algorithm is computationally attractive, and discuss its rate of convergence.

Generic uniqueness of the bias vector of finite zero-sum stochastic games with perfect information

Mean-payoff zero-sum stochastic games can be studied by means of a nonlinear spectral problem. When the state space is finite, the latter consists in finding an eigenpair (u,λ) solution of T(u)=λe+u, where T:Rn→Rn is the Shapley (or dynamic programming) operator, λ is a scalar, e is the unit vector, and u∈Rn. The scalar λ yields the mean payoff per time unit, and the vector u, called bias, allows one to determine optimal stationary strategies in the mean-payoff game. The existence of the eigenpair (u,λ) is generally related to ergodicity conditions. A basic issue is to understand for which classes of games the bias vector is unique (up to an additive constant). In this paper, we consider perfect-information zero-sum stochastic games with finite state and action spaces, thinking of the transition payments as variable parameters, transition probabilities being fixed. We show that the bias vector, thought of as a function of the transition payments, is generically unique (up to an additive constant). The proof uses techniques of nonlinear Perron–Frobenius theory. As an application of our results, we obtain an explicit perturbation scheme allowing one to solve degenerate instances of stochastic games by policy iteration.

Intra-Group Risk Sharing Under Financial Fairness

We consider the problem of finding an efficient and fair ex-ante rule for division of an uncertain monetary outcome among a finite number of von Neumann-Morgenstern agents. Efficiency is understood here, as usual, in the sense of Pareto efficiency subject to the feasibility constraint. Fairness is defined as financial fairness with respect to a predetermined pricing functional. We show that efficient and financially fair allocation rules are in one-to-one correspondence with positive eigenvectors of a nonlinear homogeneous and monotone mapping associated to the risk sharing problem. We establish relevant properties of this mapping. On the basis of this, we obtain a proof of existence and uniqueness of solutions via nonlinear Perron-Frobenius theory, as well as a proof of global convergence of the natural iterative algorithm. We argue that this algorithm is computationally attractive, and discuss its rate of convergence..

Transmit Beamforming and Power Control for Optimizing the Outage Probability Fairness in MISO Networks

This paper studies the joint beamforming and power control in a multiuser multi-input single-output network by utilizing the only statistical channel distribution information. Such information consists of slowly varying covariance matrices in the beamforming network that can be employed to reduce instantaneous feedback overhead in transmission. Utilizing solely the statistical channel information, we study how to minimize the maximum outage probability under a weighted sum power constraint that guarantees max–min fairness to all users. This problem is, however, generally hard to solve due to the nonconvexity and nonlinear coupling between beamformer and power variables. First, assuming a fixed beamformer set, we use the nonlinear Perron–Frobenius theory to design a decentralized algorithm with provable geometrically fast convergence rate to compute the optimal power. Then, for the general case, we examine a certainty-equivalent margin counterpart with outage-mapped thresholds that incorporate the statistical channel information. We show that a network duality for this certainty-equivalent problem can be useful to decouple the coupling between the beamformer and power variables. This nonlinear Perron–Frobenius theory motivated approach yields a feasible beamformer and power allocation that are near-optimal as compared to Monte Carlo averaging simulations.

Wireless Max–Min Utility Fairness With General Monotonic Constraints by Perron–Frobenius Theory

This paper presents a systematic approach for solving wireless max–min utility fairness optimization problems in multiuser wireless networks with general monotonic constraints. These problems are often challenging to solve due to their nonconvexity. By establishing a connection between this class of optimization problems and the class of conditional eigenvalue problems that can be addressed by a generalized nonlinear Perron–Frobenius theory, we show how these problems can be solved optimally using an iterative algorithm that converges geometrically fast. The mathematical development in this paper unifies previous work and allows us to handle a broader class of competitive utility functions with general nonlinear monotonic constraints. Several representative applications illustrate the effectiveness of the proposed framework, including the max–min quality-of-service subject to robust interference temperature constraints in cognitive radio networks, the min–max weighted mean-square error subject to signal-to-interference-and-noise ratio constraints in multiuser downlink systems, the max–min throughput subject to nonlinear power constraints in energyefficient wireless networks, the max–min sigmoid utility in multimedia wireless networks, and the min–max outage probability subject to outage constraints in heterogeneous wireless networks. Numerical results are presented to demonstrate the fast-convergence behavior of the algorithms to the optimal fixed-point solution characterized by our generalized nonlinear Perron–Frobenius theoretic framework.

Optimal Power Control in Rayleigh-Fading Heterogeneous Wireless Networks

Heterogeneous wireless networks provide varying degrees of network coverage in a multi-tier configuration in which low-powered small cells are used to enhance performance. Due to the ad-hoc deployment of small cells, optimal resource allocation is important to provision fairness and enhance energy efficiency. We first study the worst outage probability problem in Rayleigh-fading channels, and solve this nonconvex stochastic program using mathematical tools from nonlinear Perron-Frobenius theory. As a by-product, we solve an open problem of convergence for a previously proposed algorithm in the interference-limited case. We then address a total power minimization problem with outage specification constraints and its feasibility issue. We propose a dynamic algorithm that adapts the outage probability specification in a heterogeneous wireless network to minimize the total energy consumption and to simultaneously provide fairness guarantees in terms of the worst outage probability. Finally, we provide numerical evaluation on the performance of the algorithms and the effectiveness of deploying closed-access small cells in heterogeneous wireless networks to address the tradeoff between energy saving and feasibility of users satisfying their outage probability specifications.

Wireless Network Optimization by Perron-Frobenius Theory

A basic question in wireless networking is how to optimize the wireless network resource allocation for utility maximization and interference management. How can we overcome interference to efficiently optimize fair wireless resource allocation, under various stochastic constraints on quality of service demands? Network designs are traditionally divided into layers. How does fairness permeate through layers? Can physical layer innovation be jointly optimized with network layer routing control? How should large complex wireless networks be analyzed and designed with clearly-defined fairness using beamforming? This monograph provides a comprehensive survey of the models, algorithms, analysis, and methodologies using a Perron-Frobenius theoretic framework to solve wireless utility maximization problems. This approach overcomes the notorious non-convexity barriers in these problems, and the optimal value and solution of the optimization problems can be analytically characterized by the spectral property of matrices induced by nonlinear positive mappings. It also provides a systematic way to derive distributed and fast-convergent algorithms and to evaluate the fairness of resource allocation. This approach can even solve several previously open problems in the wireless networking literature. More generally, this approach links fundamental results in nonnegative matrix theory and linear and nonlinear Perron-Frobenius theory with the solvability of non-convex problems. In particular, for seemingly nonconvex problems, e.g., max-min wireless fairness problems, it can solve them optimally; for truly nonconvex problems, e.g., sum rate maximization, it can even be used to identify polynomial-time solvable special cases or to enable convex relaxation for global optimization.To highlight the key aspects, we also list several case studies of using the nonlinear Perron-Frobenius theoretic framework for applications in MIMO wireless cellular, heterogeneous small-cell and cognitive radio networks. Key implications arising from these work along with severalopen issues are discussed in this monograph.

Egalitarian fairness framework for joint rate and power optimization in wireless networks

How do we efficiently and fairly allocate the resource in a wireless network? We study a joint rate and power control optimization to achieve egalitarian fairness (max-min weighted fairness) in multiuser wireless networks. The key challenge to optimizing the fairness of maximizing the data rates for all the users is the nonconvexity and of the problem. We exploit the nonlinear Perron-Frobenius theory and nonnegative matrix theory to solve this nonconvex resource control problem. A fixed-point algorithm that resembles a nonlinear version of the Power Method in linear algebra and converges very fast to the optimal solution is also proposed.

Session details: Special issue on the workshop on MAthematical performance Modeling and Analysis (MAMA 2014)

The complexity of computer systems, networks and applications, as well as the advancements in computer technology, continue to grow at a rapid pace. Mathematical analysis, modeling and optimization have been playing, and continue to play, an important role in research studies to investigate fundamental issues and trade-offs at the core of performance problems in the design and implementation of complex computer systems, networks and applications. On June 20, 2014, the 16th Workshop on MAthematical performance Modeling and Analysis (MAMA 2014) was held in Austin TX, USA, sponsored by ACM SIGMETRICS, and held in conjunction with SIGMETRICS 2014. This workshop seeks to bring together researchers working on the mathematical, methodological and theoretical aspects of performance analysis, modeling and optimization. It is intended to provide a forum at SIGMETRICS conferences for talks on early research in the more mathematical areas of computer performance analysis. These talks tend to be based on very recent research results (including work in progress) or on new research results that will be otherwise submitted only to a journal (or recently have been submitted to a journal). Thus, part of the goal is to complement and supplement the SIGMETRICS Conference program with such talks without removing any theoretical contributions from the main technical program. Furthermore, we continue to experience the desired result of having abstracts from previous MAMA workshops appear as full papers in the main program of subsequent SIGMETRICS and related conferences. All submissions were reviewed by at least 4 members of the program committee, from which a total of 13 were selected for presentation at the MAMA 2014 workshop. This special issue of Performance Evaluation Review includes extended abstracts relating to these presentations (arranged in the order of their presentation), which cover a wide range of topics in the area of mathematical performance analysis, modeling and optimization. The study of Gelenbe examines the backlog of energy and of data packets in a sensor node that harvests energy, computing the properties of energy and data backlogs and discussing system stability. Meyfroyt derives asymptotic results for the coverage ratio under a specific class of spatial stochastic models (Cooperative Sequential Adsorption) and investigates the scalability of the Trickle communication protocol algorithm. The study of Tune and Roughan applies the principle of maximum entropy to develop fast traffic matrix synthesis models, with the future goal of developing realistic spatio-temporal traffic matrices. Bradonjić et al. compare and contrast the capacity, congestion and reliability requirements for alternative connectivity models of large-scale data centers relative to fat trees. The study of Rochman et al. considers the problem of resource placement in network applications, based on a largescale service faced with regionally distributed demands for various resources in cloud computing. Xie and Lui investigate the design and analysis of a rating system and a mechanism to encourage users to participate in crowdsourcing and to incentivize workers to develop high-quality solutions. The study of Asadi et al. formulates a general problem for the joint per-user mode selection, connection activation and resource scheduling of connections using both LTE and WiFi resources within the context of device-todevice communications. Zheng and Tan consider a nonconvex joint rate and power control optimization to achieve egalitarian fairness (max-min weighted fairness) in wireless networks, exploiting the nonlinear Perron-Frobenius theory and nonnegative matrix theory. The study of Goldberg et al. derives an asymptotically optimal control policy for a stochastic capacity problem of dynamically matching supply resources and uncertain demand, based on connections with lost-sales inventory models. Ghaderi et al. investigate a dynamic stochastic bin packing problem, analyzing the fluid limits of the system under an asymptotic best-fit algorithm and showing it asymptotically minimizes the number of servers used in steady state. The study of Tizghadam and Leon-Garcia examines the impact of overlaying or removing a subgraph on the Moore-Penrose inverse of the Laplacian matrix of an existing network topology and proposes an iterative method to find key performance measures. Miyazawa considers a two-node generalized Jackson network in a phase-type setting as a special case of a Markov-modulated twodimensional reflecting random walk and analyzes the tail asymptotics for this reflecting process. The study of Squillante et al. investigates improvement in scalability of search in networks through the use of multiple random walks, deriving bounds on the hitting time to a set of nodes and on various performance metrics.

Optimal Algorithms in Wireless Utility Maximization: Proportional Fairness Decomposition and Nonlinear Perron-Frobenius Theory Framework

We study the network utility maximization problems in wireless networks for service differentiation that optimize the Signal-to-Interference-plus-Noise Radio (SINR) and reliability under Rayleigh fading. Though seemingly nonconvex, we show that these problems can be decomposed into an optimization framework where each user calculates a payment for a given resource allocation, and the network uses the payment to optimize the performance of the user. We study three important examples of this utility maximization, namely the weighted sum logarithmic SINR maximization, the weighted sum inverse SINR minimization and the weighted sum logarithmic reliability maximization. These problems have hitherto been solved suboptimally in the literature. By exploiting the positivity, quasi-concavity and homogeneity properties in these problems and using the nonlinear Perron-Frobenius theory, we propose fixed-point algorithms that converge geometrically fast to the globally optimal solution. Numerical evaluations show that our algorithms are stable (free of parameter configuration) and computationally fast.

Joint Beamforming and Power Control in Coordinated Multicell: Max-Min Duality, Effective Network and Large System Transition

This paper studies joint beamforming and power control in a coordinated multicell downlink system that serves multiple users per cell to maximize the minimum weighted signal-to-interference-plus-noise ratio. The optimal solution and distributed algorithm with geometrically fast convergence rate are derived by employing the nonlinear Perron-Frobenius theory and the multicell network duality. The iterative algorithm, though operating in a distributed manner, still requires instantaneous power update within the coordinated cluster through the backhaul. The backhaul information exchange and message passing may become prohibitive with increasing number of transmit antennas and increasing number of users. In order to derive asymptotically optimal solution, random matrix theory is leveraged to design a distributed algorithm that only requires statistical information. The advantage of our approach is that there is no instantaneous power update through backhaul. Moreover, by using nonlinear Perron-Frobenius theory and random matrix theory, an effective primal network and an effective dual network are proposed to characterize and interpret the asymptotic solution.

Max-Min SINR Coordinated Multipoint Downlink Transmission—Duality and Algorithms

This paper considers the max-min weighted signal-to-interference-plus-noise ratio (SINR) problem subject to multiple weighted-sum power constraints, where the weights can represent relative power costs of serving different users. First, we study the power control problem. We apply nonlinear Perron-Frobenius theory to derive closed-form expressions for the optimal value and solution and an iterative algorithm which converges geometrically fast to the optimal solution. Then, we use the structure of the closed-form solution to show that the problem can be decoupled into subproblems each involving only one power constraint. Next, we study the multiple-input-single-output (MISO) transmit beamforming and power control problem. We use uplink-downlink duality to show that this problem can be decoupled into subproblems each involving only one power constraint. We apply this decoupling result to derive an iterative subgradient projection algorithm for the problem.

TROPICAL POLYHEDRA ARE EQUIVALENT TO MEAN PAYOFF GAMES

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron–Frobenius theory.