Optimal distributed linear averaging

A new optimal distributed linear averaging (ODLA) problem is presented in this paper. This problem is motivated by the distributed averaging problem which arises in the context of distributed algorithms in computer science and coordination of groups of autonomous agents in engineering. The aim of the ODLA problem is to compute the average of the initial values at nodes of a graph through an optimal distributed algorithm in which the nodes in the graph can only communicate with their neighbors. Optimality is given by a minimization problem of a quadratic cost functional under infinite horizon. We show that this problem has a very close relationship with the notion of semistability. By developing new necessary and sufficient conditions for semistability of linear discrete-time systems, we convert the proposed ODLA problem into an equivalent, constrained optimization problem and then derive a solvable, fixed-structure convex optimization problem.     

Fast linear iterations for distributed averaging

We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging linear iteration can be cast as a semidefinite program, and therefore efficiently and globally solved. These optimal linear iterations are often substantially faster than several common heuristics that are based on the Laplacian of the associated graph.We show how problem structure can be exploited to speed up interior-point methods for solving the fastest distributed linear iteration problem, for networks with up to a thousand or so edges. We also describe a simple subgradient method that handles far larger problems, with up to 100 000 edges. We give several extensions and variations on the basic problem.     

Distributed control over structured and packet‐dropping networks

In this paper we consider distributed control of n dynamic agents to optimize an overall system performance metric. Due to limited communication resources, there exist structured interconnections among the agents and the interest is placed on synthesizing a suitably distributed control law to provide a given performance level. Based on a Youla–Kucera (Y–K) parameterization approach, the problem of designing a distributed controller to deliver given performance levels for different network topologies is shown to be convex in the Y–K parameter Q. Furthermore, if in addition to structured interconnections, packet drops exist in information transmission among the agents, we provide convex conditions to guarantee mean square (MS) stability and to optimize ℋ︁₂ system performance. The proposed method is also extended to deal with systems of triangular structure. Copyright © 2007 John Wiley & Sons, Ltd.     

Orthogonal linear regression algorithm based on augmented matrix formulation

The problem of n-dimensional orthogonal linear regression is a problem of finding an n-dimensional hyperplane minimising the sum of Euclidean distances between this hyperplane and a given set of m points, where m n. This nonlinear programming problem has been re-cast in an augmented matrix form and solved as a sequence of iteratively re-weighted least square problems. The proposed algorithm is seen as an alternative to the recently published algorithm by Cavalier and Melloy [1].     

Optimum parameter for the SOR-like method for augmented systems

Recently, several proposals for the generalization of Young's SOR method to the saddle point problem or the augmented system has been presented. One of the most practical versions is the SOR-like method given by Golub et al., [(2001). SOR-like methods for augmented systems. BIT, 41, 71–85.], where the convergence and the determination of its optimum parameters were given. In this article, a full characterization of the spectral radius of the SOR-like iteration matrix is given, and an explicit expression for the optimum parameter is given in each case. The new results also lead to different results to that of Golub et al. Besides, it is shown that by the choices of the preconditioning matrix, the optimum SOR-like iteration matrix has no complex eigenvalues, therefore, it can be accelerated by semi-iterative methods.     

Distributed robust control of linear multi-agent systems with parameter uncertainties

This article considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighbouring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the H _∞ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust H _∞ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.     

An Edge-based Stochastic Proximal Gradient Algorithm for Decentralized Composite Optimization

This paper investigates decentralized composite optimization problems involving a common non-smooth regularization term over an undirected and connected network. In the same situation, there exist lots of gradient-based proximal distributed methods, but most of them are only sublinearly convergent. The proof of linear convergence for this series of algorithms is extremely difficult. To set up the problem, we presume all networked agents use the same non-smooth regularization term, which is the circumstance for most machine learning to implement based on centralized optimization. For this scenario, most existing proximal-gradient algorithms trend to ignore the cost of gradient evaluations, which results in degraded performance. To tackle this problem, we further set the local cost function to the average of a moderate amount of local cost subfunctions and develop an edge-based stochastic proximal gradient algorithm (SPG-Edge) by employing local unbiased stochastic averaging gradient method. When the non-smooth term does not exist, the proposed algorithm could be extended to some notable primal-dual domain algorithms, such as EXTRA and DIGing. Finally, we provide a simplified proof of linear convergence and conduct numerical experiments to illustrate the validity of theoretical results.

     

Distributed robust control of uncertain linear multi‐agent systems

In this paper, the distributed H _∞ robust control problem synthesized with transient performance is investigated for a group of autonomous agents governed by uncertain general linear node dynamics. Based on the relative information between neighboring agents and some information of other agents, distributed state‐feedback and observer‐type output‐feedback control protocols are designed and analyzed, respectively. By using tools from robust control theory, conditions for the existence of controllers for solving such a problem are established. It is shown that the problem of distributed H _∞ robust control synthesized with transient performance can be converted to the H _∞ control problem synthesized with transient performance for decoupled linear systems of the same low dimensions. Finally, simulation examples are provided to illustrate the effectiveness of the design. Copyright © 2014 John Wiley & Sons, Ltd.     

Multigrid methods for indefinite Toeplitz matrices

In this paper we modify a multigrid technique introduced for illconditioned symmetric positive definite Toepliltz linear systems [22] in order to deal with the nondefinite case. This kind of linear systems arises in important applications such as image restoration, harmonic retrieval etc. We prove that the cost per iteration isO(nlog² n) ops andO(log² n) parallel steps while the convergence speed seems to be independent of the problem matrix-size. Therefore this technique results to be competitive with respect to the fast direct method devised by Chan and Hansen [16] and is alternative to the PCG-based algorithm introduced in [33, 32, 37].     

High‐order internal model‐based iterative learning control design for nonlinear distributed parameter systems

This article deals with the problem of iterative learning control algorithm for a class of nonlinear parabolic distributed parameter systems (DPSs) with iteration‐varying desired trajectories. Here, the variation of the desired trajectories in the iteration domain is described by a high‐order internal model. According to the characteristics of the systems, the high‐order internal model‐based P‐type learning algorithm is constructed for such nonlinear DPSs, and furthermore, the corresponding convergence theorem of the presented algorithm is established. It is shown that the output trajectory can converge to the desired trajectory in the sense of (L²,λ) ‐norm along the iteration axis within arbitrarily small error. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.     

Designing linear distributed algorithms with memory for fast convergence

Motivated by both distributed computation and decentralized control applications, we studied the distributed linear iterative algorithms with memory. Specifically, we showed that the system of linear equations G x = b can be solved through a distributed linear iteration for arbitrary invertible G using only a single memory element at each processor. Further, we demonstrated that the memoried distributed algorithm can be designed to achieve much faster convergence than a memoryless distributed algorithm. Two small simulation examples were included to illustrate the results. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic switched controller design of networked control systems with a random long delay

The problem of stabilizing Networked Control Systems (NCSs) with random but bounded delays is discussed in this paper. By using an augmented state‐space method, this class of problem can be modeled as a discrete‐time jump linear system governed by finite‐state Markov chains. As the network‐induced time vary delay of NCSs changes along with the network transferring route and the network load, results in systems becoming instable with controller designs based on a fixed transition matrix, we firstly make use of the V‐K iteration algorithm to design m groups of stabilizing controllers that satisfy different m transition matrixes, and then constitute a switched controller for them and a switch. The simulation shows that if the switched controller is used to stabilize the discrete‐time jump linear system, this system not only has a larger stabilizing span, but also has better dynamic stabilizing characters compared to those with only one group of controllers. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Lower bounds for minimizing total completion time in a two-machine flow shop

For the -hard problem of scheduling n jobs in a two-machine flow shop so as to minimize the total completion time, we present two equivalent lower bounds that are computable in polynomial time. We formulate the problem by the use of positional completion time variables, which results in two integer linear programming formulations with O(n ²) variables and O(n) constraints. Solving the linear programming relaxation renders a very strong lower bound with an average relative gap of only 0.8% for instances with more than 30 jobs. We further show that relaxing the formulation in terms of positional completion times by applying Lagrangean relaxation yields the same bound, no matter which set of constraints we relax.     

基于矩阵和求逆及MRC的低复杂度MIMO检测器

信道编码MIMO系统需要检测器具有软输入软输出特性,而常规的检测算法通常具有很高的计算复杂度,阻碍了其在实际中的应用.提出一种低复杂度MIMO检测方案.首次迭代中,利用低复杂度快速矩阵和分解方案来获得MMSE检测输出,避免了常规矩阵和求逆中的Jordan标准型化简;其余迭代中,利用信道解码器提供的软信息将MIMO系统转...     

A class of iterative methods for computing polar decomposition

In this article, there is offered a parametric class of iterative methods for computing the polar decomposition of a matrix. Each iteration of this class needs only one scalar-by-matrix and three matrix-by-matrix multiplications. It is no use computing inversion, so no numerical problems can be created because of ill-conditioning. Some available methods can be included in this class by choosing a suitable value for the parameter. There are obtained conditions under which this class is always quadratically convergent. The numerical comparison performed among six quadratically convergent methods for computing polar decomposition, and a special method of this class, chosen based on a specific value for the parameter, shows that the number of iterations of the special method is considerably near that of a cubically convergent Halley's method. Ten n×n matrices with n=5, 10, 20, 50, 100 were chosen to make this comparison.     

Minimum polygonal separation

In this paper we study the problem of polygonal separation in the plane, i.e., finding a convex polygon with minimum number k of sides separating two given finite point sets (k-separator), if it exists. We show that for k = Θ(n), Ω(n log n) is a lower bound to the running time of any algorithm for this problem, and exhibit two algorithms of distinctly different flavors. The first relies on an O(n log n)-time preprocessing task, which constructs the convex hull of the internal set and a nested star-shaped polygon determined by the external set; the k-separator is contained in the annulus between the boundaries of these two polygons and is constructed in additional linear time. The second algorithm adapts the prune-and-search approach, and constructs, in each iteration, one side of the separator; its running time is O(kn), but the separator may have one more side than the minimum.     

Lie- and chronologico-algebraic tools for studying stability of time-varying systems

We will study stability and asymptotic stability for time-varying systems described by ODEs of the form , where f(t,x) is 1-periodic with respect to t and >0 is a small parameter. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there have been a lot of study regarding stability of such systems and design of fast-oscillating stabilizing feedback laws. In this paper we suggest an approach which is kind of high-order averaging procedure based on Lie algebraic formalism and the formalism of chronological calculus. This latter is a method of asymptotic analysis for flows generated by time-variant ODE. We apply the approach to study stability issues for linear and nonlinear systems. In particular, we derive conditions of stability for the second- and third-order linear differential equations with periodic fast-oscillating coefficients, we study output-feedback stabilization of bilinear systems and consider high-order averaging procedure for nonlinear systems under homogeneity assumptions. At the end we study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks.     

Fixed-Parameter Complexity of Minimum Profile Problems

The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n−1+k, considering k as the parameter; this is a parameterization above guaranteed value, since n−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. A preliminary version of the paper is published in Proc. IWPEC 2006, LNCS vol. 4169, 60–71.     

Recurrent averaging inequalities in multi-agent control and social dynamics modeling

Many multi-agent control algorithms and dynamic agent-based models arising in natural and social sciences are based on the principle of iterative averaging. Each agent is associated to a value of interest, which may represent, for instance, the opinion of an individual in a social group, the velocity vector of a mobile robot in a flock, or the measurement of a sensor within a sensor network. This value is updated, at each iteration, to a weighted average of itself and of the values of the adjacent agents. It is well known that, under natural assumptions on the network’s graph connectivity, this local averaging procedure eventually leads to global consensus, or synchronization of the values at all nodes. Applications of iterative averaging include, but are not limited to, algorithms for distributed optimization, for solution of linear and nonlinear equations, for multi-robot coordination and for opinion formation in social groups. Although these algorithms have similar structures, the mathematical techniques used for their analysis are diverse, and conditions for their convergence and differ from case to case. In this paper, we review many of these algorithms and we show that their properties can be analyzed in a unified way by using a novel tool based on recurrent averaging inequalities (RAIs). We develop a theory of RAIs and apply it to the analysis of several important multi-agent algorithms recently proposed in the literature.     

Analysis of Randomized Protocols for Conflict-Free Distributed Access

We study the following distributed access problem which arises naturally in many settings: given a set of n data items shared among n nodes in a distributed network, all nodes want to access all (or a subset of) the items residing on different nodes in a conflict-free manner. In addition, items may move from one node to the other during access. Our goal is to design distributed protocols so that all nodes access all the desired items as quickly as possible, while at the same time not overloading the storage space of any one node. Using centralized coordination among the nodes it is easy to design an optimal scheme in which all nodes can access all the items in n−1 steps storing only one item at any time. We show that a simple randomized distributed protocol performs almost as well as the optimal (centralized) scheme but with no coordination overhead. Our protocol takes O(n) time with high probability to access all n items which is asymptotically as good as the optimal centralized scheme. The protocol guarantees that the maximum load (the maximum number of items stored in any node) at any time is at most O(log n/log log n) with high probability which is only slightly larger compared to the Ω(1) load of the optimal scheme. Our analysis involves a stochastic analysis of a “balls into bins” problem in a dynamic setting where balls (data items) move into bins (nodes) on request and we study the time and load requirements to move all the balls to the requested bins. A short version of this paper appeared in the Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing (PODC), 2005.