distributed nonconvex constrained optimization over time varying digraphs this paper considers nonconvex distributed constrained optimization over networks modeled as directed possibly time varying graphs we introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex nonseparable function the agent s sum utility plus a difference of convex function with nonsmooth convex part this general formulation arises in many applications from statistical machine learning to engineering the proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push sum consensus mechanism that aims to track locally the gradient of the smooth part of the sum utility sublinear convergence rate is proved when a fixed step size possibly different among the agents is employed whereas asymptotic convergence to stationary solutions is proved using a diminishing step size numerical results show that our algorithms compare favorably with current schemes on both convex and nonconvex problems

dqm decentralized quadratically approximated alternating direction method of multipliers this paper considers decentralized consensus optimization problems where nodes of a network have access to different summands of a global objective function nodes cooperate to minimize the global objective by exchanging information with neighbors only a decentralized version of the alternating directions method of multipliers dadmm is a common method for solving this category of problems dadmm exhibits linear convergence rate to the optimal objective for strongly convex functions but its implementation requires solving a convex optimization problem at each iteration this can be computationally costly and may result in large overall convergence times the decentralized quadratically approximated admm algorithm dqm which minimizes a quadratic approximation of the objective function that dadmm minimizes at each iteration is proposed here the consequent reduction in computational time is shown to have minimal effect on convergence properties convergence still proceeds at a linear rate with a guaranteed factor that is asymptotically equivalent to the dadmm linear convergence rate factor numerical results demonstrate advantages of dqm relative to dadmm and other alternatives in a logistic regression problem

communication efficient algorithms for decentralized and stochastic optimization we present a new class of decentralized first order methods for nonsmooth and stochastic optimization problems defined over multiagent networks considering that communication is a major bottleneck in decentralized optimization our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter node communications our major contribution is to present a new class of decentralized primal dual type algorithms namely the decentralized communication sliding dcs methods which can skip the inter node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions by employing dcs agents can find an epsilon solution both in terms of functional optimality gap and feasibility residual in mathcal o 1 epsilon resp mathcal o 1 sqrt epsilon communication rounds for general convex functions resp strongly convex functions while maintaining the mathcal o 1 epsilon 2 resp mathcal o 1 epsilon bound on the total number of intra node subgradient evaluations we also present a stochastic counterpart for these algorithms denoted by sdcs for solving stochastic optimization problems whose objective function cannot be evaluated exactly in comparison with existing results for decentralized nonsmooth and stochastic optimization we can reduce the total number of inter node communication rounds by orders of magnitude while still maintaining the optimal complexity bounds on intra node stochastic subgradient evaluations the bounds on the stochastic subgradient evaluations are actually comparable to those required for centralized nonsmooth and stochastic optimization under certain conditions on the target accuracy

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

on the linear convergence of the admm in decentralized consensus optimization in decentralized consensus optimization a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable where their information exchange is restricted between the neighbors to this end one can first obtain a problem reformulation and then apply the alternating direction method of multipliers admm the method applies iterative computation at the individual agents and information exchange between the neighbors this approach has been observed to converge quickly and deemed powerful this paper establishes its linear convergence rate for the decentralized consensus optimization problem with strongly convex local objective functions the theoretical convergence rate is explicitly given in terms of the network topology the properties of local objective functions and the algorithm parameter this result is not only a performance guarantee but also a guideline toward accelerating the admm convergence

further comments on the one dimensional percolation problem with multineighbour bonds one dimensional site and bond percolation problems with bonds connecting lth nearest neighbours are solved exactly by using an infinite cell to cell renormalisation group transformation both thermal and magnetic critical exponents are obtained through the scaling relations although universality breaks down for site and bond percolation when l or 2 suzuki s weaker universality 1974 still holds

decentralized maximum likelihood estimation for sensor networks composed of nonlinearly coupled dynamical systems in this paper we propose a decentralized sensor network scheme capable to reach a globally optimum maximum likelihood ml estimate through self synchronization of nonlinearly coupled dynamical systems each node of the network is composed of a sensor and a first order dynamical system initialized with the local measurements nearby nodes interact with each other exchanging their state value and the final estimate is associated to the state derivative of each dynamical system we derive the conditions on the coupling mechanism guaranteeing that if the network observes one common phenomenon each node converges to the globally optimal ml estimate we prove that the synchronized state is globally asymptotically stable if the coupling strength exceeds a given threshold acting on a single parameter the coupling strength we show how in the case of nonlinear coupling the network behavior can switch from a global consensus system to a spatial clustering system finally we show the effect of the network topology on the scalability properties of the network and we validate our theoretical findings with simulation results

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

diland an algorithm for distributed sensor localization with noisy distance measurements we present an algorithm for distributed sensor localization with noisy distance measurements diland that extends and makes the dlre more robust dlre is a distributed sensor localization algorithm in rm m 1 introduced in our previous work ieee trans signal process vol 57 no 5 pp 2000 2016 may 2009 diland operates when 1 the communication among the sensors is noisy 2 the communication links in the network may fail with a nonzero probability and 3 the measurements performed to compute distances among the sensors are corrupted with noise the sensors which do not know their locations lie in the convex hull of at least m 1 anchors nodes that know their own locations under minimal assumptions on the connectivity and triangulation of each sensor in the network we show that under the broad random phenomena described above diland converges almost surely a s to the exact sensor locations

long range percolation in one dimension the problem of long range percolation in one dimension is proposed the authors consider a one dimensional bond percolation system with bonds connecting an infinite number of neighbours where the occupation probability for the nth nearest neighbour bond pn varies as p1 ns using the transfer matrix method they find that when s 2 only the short range percolation exists namely the system percolates only when p1 1 a transition to long range percolation is found at s 2 where the percolation threshold drops suddenly from the short range value p1c 1 to the long range value p1c 0

global existence of radial solutions for general semilinear hyperbolic systems in 3d abstract we study the well posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions we give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3d under null condition with small scaling invariant w 2 1 r 3 data we obtain a bilinear estimate that is effective for hyperbolic systems which do not have any time decay it allows us to achieve the boundedness of the weighted bv norm of the radial solutions

dlm decentralized linearized alternating direction method of multipliers this paper develops the decentralized linearized alternating direction method of multipliers dlm that minimizes a sum of local cost functions in a multiagent network the algorithm mimics operation of the decentralized alternating direction method of multipliers dadmm except that it linearizes the optimization objective at each iteration this results in iterations that instead of successive minimizations implement steps whose cost is akin to the much lower cost of the gradient descent step used in the distributed gradient method dgm the algorithm is proven to converge to the optimal solution when the local cost functions have lipschitz continuous gradients its rate of convergence is shown to be linear if the local cost functions are further assumed to be strongly convex numerical experiments in least squares and logistic regression problems show that the number of iterations to achieve equivalent optimality gaps are similar for dlm and admm and both much smaller than those of dgm in that sense dlm combines the rapid convergence of admm with the low computational burden of dgm

comments on the transfer matrix approach to the one dimensional bond percolation with further neighbour bonds the transfer matrices which have been used in previous studies on the one dimensional bond percolation problem with bonds connecting lth nearest neighbours are only for the case of directed bonds for the case of undirected bonds the transfer matrices are non systematic and become much more complicated when l or 3 here the authors study the cases of l 2 3 and find that the critical behaviour remains unchanged they strongly suggest that this is true for any l rigorous evidence is given to support this conjecture

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

explicit convergence rate of a distributed alternating direction method of multipliers consider a set of n agents seeking to solve distributively the minimization problem inf x sum n 1 n f n x where the convex functions f n are local to the agents the popular alternating direction method of multipliers has the potential to handle distributed optimization problems of this kind we provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures the rate of convergence of our method is considered it is assumed that the infimum of the problem is reached at a point x star the functions f n are twice differentiable at this point and sum nabla 2 f n x star 0 in the positive definite ordering of symmetric matrices with these assumptions it is shown that the convergence to the consensus x star is linear and the exact rate is provided application examples where this rate can be optimized with respect to the admm free parameter rho are also given

optimal parameter selection for the alternating direction method of multipliers admm quadratic problems the alternating direction method of multipliers admm has emerged as a powerful technique for large scale structured optimization despite many recent results on the convergence properties of admm a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking in this paper we find the optimal algorithm parameters that minimize the convergence factor of the admm iterates in the context of l 2 regularized minimization and constrained quadratic programming numerical examples show that our parameter selection rules significantly outperform existing alternatives in the literature

blowup phenomena of solutions to the euler equations for compressible fluid flow abstract the blowup phenomena of solutions is investigated for the euler equations of compressible fluid flow the approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi dimensional solutions in particular the special solutions with velocity of the form c t x are constructed to show the expanding and blowup properties the solution with velocity of the form a t x a t for 1 and for any space dimensions is obtained as a corollary another conclusion is that there is only trivial solution with velocity of the form c t x 1 x for 1 and multi space dimensions

linear convergence rate of a class of distributed augmented lagrangian algorithms we study distributed optimization where nodes cooperatively minimize the sum of their individual locally known convex costs f i x s x in bbr d is global distributed augmented lagrangian al methods have good empirical performance on several signal processing and learning applications but there is limited understanding of their convergence rates and how it depends on the underlying network this paper establishes globally linear geometric convergence rates of a class of deterministic and randomized distributed al methods when the f i s are twice continuously differentiable and have a bounded hessian we give explicit dependence of the convergence rates on the underlying network parameters simulations illustrate our analytical findings

fast consensus by the alternating direction multipliers method the alternating direction multipliers method admm has been recently proposed as a practical and efficient algorithm for distributed computing we discuss its applicability to the average consensus problem in this paper by carefully relaxing admm augmentation coefficients we are able to analytically investigate its properties and to propose simple and strict analytical bounds these provide a clear indication on how to choose system parameters for optimized performance we prove both analytically and via simulations that the proposed approach exhibits convergence speed between the best in the literature classical and optimized solutions while providing the most powerful resilience to noise

one dimensional percolation problems with further neighbour bonds transfer matrix approach the transfer matrix method is used to find the exact expressions for the correlation length in the critical region for both one dimensional site and bond percolation problems with bonds connecting lth nearest neighbours for any finite l for the site percolation the correlation length exponent v is found to be l consistent with the result obtained from the generating function method for the bond percolation we find v l l 1 2

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

fast distributed gradient methods we study distributed optimization problems when n nodes minimize the sum of their individual costs subject to a common vector variable the costs are convex have lipschitz continuous gradient with constant l and bounded gradient we propose two fast distributed gradient algorithms based on the centralized nesterov gradient algorithm and establish their convergence rates in terms of the per node communications k and the per node gradient evaluations k our first method distributed nesterov gradient achieves rates o logk k and o logk k our second method distributed nesterov gradient with consensus iterations assumes at all nodes knowledge of l and w the second largest singular value of the n n doubly stochastic weight matrix w it achieves rates o 1 k2 and o 1 k2 0 arbitrarily small further we give for both methods explicit dependence of the convergence constants on n and w simulation examples illustrate our findings

decomposition by partial linearization parallel optimization of multi agent systems we propose a novel decomposition framework for the distributed optimization of general nonconvex sum utility functions arising naturally in the system design of wireless multi user interfering systems our main contributions are i the development of the first class of inexact jacobi best response algorithms with provable convergence where all the users simultaneously and iteratively solve a suitably convexified version of the original sum utility optimization problem ii the derivation of a general dynamic pricing mechanism that provides a unified view of existing pricing schemes that are based instead on heuristics and iii a framework that can be easily particularized to well known applications giving rise to very efficient practical jacobi or gauss seidel algorithms that outperform existing ad hoc methods proposed for very specific problems interestingly our framework contains as special cases well known gradient algorithms for nonconvex sum utility problems and many block coordinate descent schemes for convex functions

consensus in ad hoc wsns with noisy links part i distributed estimation of deterministic signals we deal with distributed estimation of deterministic vector parameters using ad hoc wireless sensor networks wsns we cast the decentralized estimation problem as the solution of multiple constrained convex optimization subproblems using the method of multipliers in conjunction with a block coordinate descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation different from existing alternatives our approach does not require the centralized estimator to be expressible in a separable closed form in terms of averages thus allowing for decentralized computation even of nonlinear estimators including maximum likelihood estimators mle in nonlinear and non gaussian data models we prove that these algorithms have guaranteed convergence to the desired estimator when the sensor links are assumed ideal furthermore our decentralized algorithms exhibit resilience in the presence of receiver and or quantization noise in particular we introduce a decentralized scheme for least squares and best linear unbiased estimation blue and establish its convergence in the presence of communication noise our algorithms also exhibit potential for higher convergence rate with respect to existing schemes corroborating simulations demonstrate the merits of the novel distributed estimation algorithms

transfer matrix approach to the one dimensional percolation problem the transfer matrix method is used to find the exact phase diagrams and the correlation length exponents v for the one dimensional site bond and site bond percolation problems with bonds connecting the lth nearest neighbours l up to 3 for the site percolation the authors results agree with the exact result obtained from the generating function method v l while for the bond percolation they found much richer critical phenomena if all the l bonds have equal occupation probability their results predict v l l 1 2