Showing papers on "Convergence (routing) published in 2013"

A unified convergence analysis of block successive minimization methods for nonsmooth optimization

Abstract: The block coordinate descent (BCD) method is widely used for minimizing a continuous function $f$ of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of $f$ which are either locally tight upper bounds of $f$ or strictly convex local approximations of $f$. The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results ...

Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm for Non-Convex Optimization

Abstract: We introduce a new framework for the convergence analysis of a class of distributed constrained non-convex optimization algorithms in multi-agent systems. The aim is to search for local minimizers of a non-convex objective function which is supposed to be a sum of local utility functions of the agents. The algorithm under study consists of two steps: a local stochastic gradient descent at each agent and a gossip step that drives the network of agents to a consensus. Under the assumption of decreasing stepsize, it is proved that consensus is asymptotically achieved in the network and that the algorithm converges to the set of Karush-Kuhn-Tucker points. As an important feature, the algorithm does not require the double-stochasticity of the gossip matrices. It is in particular suitable for use in a natural broadcast scenario for which no feedback messages between agents are required. It is proved that our results also holds if the number of communications in the network per unit of time vanishes at moderate speed as time increases, allowing potential savings of the network's energy. Applications to power allocation in wireless ad-hoc networks are discussed. Finally, we provide numerical results which sustain our claims.

Accelerating a Recurrent Neural Network to Finite-Time Convergence for Solving Time-Varying Sylvester Equation by Using a Sign-Bi-power Activation Function

Abstract: Bartels---Stewart algorithm is an effective and widely used method with an O(n 3) time complexity for solving a static Sylvester equation. When applied to time-varying Sylvester equation, the computation burden increases intensively with the decrease of sampling period and cannot satisfy continuous realtime calculation requirements. Gradient-based recurrent neural network are able to solve the time-varying Sylvester equation in real time but there always exists an estimation error. In contrast, the recently proposed Zhang neural network has been proven to converge to the solution of the Sylvester equation ideally when time goes to infinity. However, this neural network with the suggested activation functions never converges to the desired value in finite time, which may limit its applications in realtime processing. To tackle this problem, a sign-bi-power activation function is proposed in this paper to accelerate Zhang neural network to finite-time convergence. The global convergence and finite-time convergence property are proven in theory. The upper bound of the convergence time is derived analytically. Simulations are performed to evaluate the performance of the neural network with the proposed activation function. In addition, the proposed strategy is applied to online calculating the pseudo-inverse of a matrix and nonlinear control of an inverted pendulum system. Both theoretical analysis and numerical simulations validate the effectiveness of proposed activation function.

On the O(1/k) Convergence of Asynchronous Distributed Alternating Direction Method of Multipliers

Abstract: We consider a network of agents that are cooperatively solving a global optimization problem, where the objective function is the sum of privately known local objective functions of the agents and the decision variables are coupled via linear constraints. Recent literature focused on special cases of this formulation and studied their distributed solution through either subgradient based methods with O(1/sqrt(k)) rate of convergence (where k is the iteration number) or Alternating Direction Method of Multipliers (ADMM) based methods, which require a synchronous implementation and a globally known order on the agents. In this paper, we present a novel asynchronous ADMM based distributed method for the general formulation and show that it converges at the rate O(1/k).

Finite-Horizon Control-Constrained Nonlinear Optimal Control Using Single Network Adaptive Critics

Abstract: To synthesize fixed-final-time control-constrained optimal controllers for discrete-time nonlinear control-affine systems, a single neural network (NN)-based controller called the Finite-horizon Single Network Adaptive Critic is developed in this paper. Inputs to the NN are the current system states and the time-to-go, and the network outputs are the costates that are used to compute optimal feedback control. Control constraints are handled through a nonquadratic cost function. Convergence proofs of: 1) the reinforcement learning-based training method to the optimal solution; 2) the training error; and 3) the network weights are provided. The resulting controller is shown to solve the associated time-varying Hamilton-Jacobi-Bellman equation and provide the fixed-final-time optimal solution. Performance of the new synthesis technique is demonstrated through different examples including an attitude control problem wherein a rigid spacecraft performs a finite-time attitude maneuver subject to control bounds. The new formulation has great potential for implementation since it consists of only one NN with single set of weights and it provides comprehensive feedback solutions online, though it is trained offline.

A Distributed Newton Method for Network Utility Maximization–I: Algorithm

Abstract: Most existing works use dual decomposition and first-order methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This paper develops an alternative distributed Newton-type fast converging algorithm for solving NUM problems. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner. We propose a stepsize rule and provide a distributed procedure to compute it in finitely many iterations. The key feature of our direction and stepsize computation schemes is that both are implemented using the same distributed information exchange mechanism employed by first order methods. We describe the details of the inexact algorithm here and in part II of this paper , we show that under some assumptions, even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly in terms of primal iterations to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing first-order methods based on dual decomposition.

A new tool to study real dynamics: The Convergence Plane

Abstract: In this paper, the author presents a new tool, called The Convergence Plane, that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool can be used, inter alia, to find the elements of a family that have good convergence properties and discard the bad ones or to see how the basins of attraction changes along the elements of the family. To show the applicability of the tool an example of the dynamics of the Damped Newton's method applied to a cubic polynomial is presented.

Accelerated and Inexact Forward-Backward Algorithms

Abstract: We propose a convergence analysis of accelerated forward-backward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the $1/k^2$ convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented.

On Convergence Time and Disturbance Rejection of Super-Twisting Control

Abstract: Super-twisting algorithm is one of the versions of high-order sliding mode control. The interest to this algorithm is explained by its attractive properties: continuous control input, finite convergence time, disturbance rejection. In this paper, the upper bound of admissible unknown disturbances and low bound of the convergence time are found and shown that both the values can be achieved with any desired accuracy.

Local Linear Convergence of the Alternating Direction Method of Multipliers on Quadratic or Linear Programs

Abstract: We introduce a novel matrix recurrence yielding a new spectral analysis of the local transient convergence behavior of the alternating direction method of multipliers (ADMM), for the particular case of a quadratic program or a linear program. We identify a particular combination of vector iterates whose convergence can be analyzed via a spectral analysis. The theory predicts that ADMM should go through up to four convergence regimes, such as constant step convergence or linear convergence, ending with the latter when close enough to the optimal solution if the optimal solution is unique and satisfies strict complementarity.

Unconditional convergence and optimal error estimates of a galerkin-mixed fem for incompressible miscible flow in porous media ∗

Abstract: In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Our theoretical results provide a new understanding on commonly used linearized schemes. The proof is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

Dynamic Anomalography: Tracking Network Anomalies Via Sparsity and Low Rank

Abstract: In the backbone of large-scale networks, origin-to-destination (OD) traffic flows experience abrupt unusual changes known as traffic volume anomalies, which can result in congestion and limit the extent to which end-user quality of service requirements are met. As a means of maintaining seamless end-user experience in dynamic environments, as well as for ensuring network security, this paper deals with a crucial network monitoring task termed dynamic anomalography. Given link traffic measurements (noisy superpositions of unobserved OD flows) periodically acquired by backbone routers, the goal is to construct an estimated map of anomalies in real time, and thus summarize the network `health state' along both the flow and time dimensions. Leveraging the low intrinsic-dimensionality of OD flows and the sparse nature of anomalies, a novel online estimator is proposed based on an exponentially-weighted least-squares criterion regularized with the sparsity-promoting l1-norm of the anomalies, and the nuclear norm of the nominal traffic matrix. After recasting the non-separable nuclear norm into a form amenable to online optimization, a real-time algorithm for dynamic anomalography is developed and its convergence established under simplifying technical assumptions. For operational conditions where computational complexity reductions are at a premium, a lightweight stochastic gradient algorithm based on Nesterov's acceleration technique is developed as well. Comprehensive numerical tests with both synthetic and real network data corroborate the effectiveness of the proposed online algorithms and their tracking capabilities, and demonstrate that they outperform state-of-the-art approaches developed to diagnose traffic anomalies.

On convergence of homotopy analysis method and its application to fractional integro-differential equations

Abstract: In this paper, we have used the homotopy analysis method (HAM) to obtain approximate solution of fractional integro-differential equations (FIDEs). Convergence of HAM is considered for this kind of equations. Also some examples are given to illustrate the high efficiency and precision of HAM. Keywords: Fractional integro-differential equation, homotopy analysis method, convergence control parameter Quaestiones Mathematicae 36(2013), 93–105

Adaptive Primal-Dual Hybrid Gradient Methods for Saddle-Point Problems

Abstract: The Primal-Dual hybrid gradient (PDHG) method is a powerful optimization scheme that breaks complex problems into simple sub-steps. Unfortunately, PDHG methods require the user to choose stepsize parameters, and the speed of convergence is highly sensitive to this choice. We introduce new adaptive PDHG schemes that automatically tune the stepsize parameters for fast convergence without user inputs. We prove rigorous convergence results for our methods, and identify the conditions required for convergence. We also develop practical implementations of adaptive schemes that formally satisfy the convergence requirements. Numerical experiments show that adaptive PDHG methods have advantages over non-adaptive implementations in terms of both efficiency and simplicity for the user.

A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications

Abstract: A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDEs) in which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. The proposed special balanced scheme is explicit and its mean-square order of convergence is 1/2. Some numerical tests are presented.

Convergence Properties of a Gradual Learning Algorithm for Harmonic Grammar

Abstract: This chapter investigates a gradual on-line learning algorithm for Harmonic Grammar. By adapting existing convergence proofs for perceptrons, we show that for any nonvarying target language, Harmonic-Grammar learners are guaranteed to converge to an appropriate grammar, if they receive complete information about the structure of the learning data. We also prove convergence when the learner incorporates evaluation noise, as in Stochastic Optimality Theory. Computational tests of the algorithm show that it converges quickly. When learners receive incomplete information (e.g. some structure remains hidden), tests indicate that the algorithm is more likely to converge than two comparable Optimality-Theoretic learning algorithms.

On fixed and finite time stability in sliding mode control

Abstract: Any finite-time convergent homogeneous sliding-mode controller can be transformed into a fixed-time convergent one, featuring an upper bound of convergence time, which does not depend on initial conditions. Output feedback controller is optional. Continuity of the convergence time functions of homogeneous differential inclusions is studied. Feasibility of fixed-time-stable systems is considered.

Robust Distributed Routing in Dynamical Networks—Part I: Locally Responsive Policies and Weak Resilience

Abstract: Robustness of distributed routing policies is studied for dynamical networks, with respect to adversarial disturbances that reduce the link flow capacities. A dynamical network is modeled as a system of ordinary differential equations derived from mass conservation laws on a directed acyclic graph with a single origin-destination pair and a constant total outflow at the origin. Routing policies regulate the way the total outflow at each nondestination node gets split among its outgoing links as a function of the current particle density, while the outflow of a link is modeled to depend on the current particle density on that link through a flow function. The dynamical network is called partially transferring if the total inflow at the destination node is asymptotically bounded away from zero, and its weak resilience is measured as the minimum sum of the link-wise magnitude of disturbances that make it not partially transferring. The weak resilience of a dynamical network with arbitrary routing policy is shown to be upper bounded by the network's min-cut capacity and, hence, is independent of the initial flow conditions. Moreover, a class of distributed routing policies that rely exclusively on local information on the particle densities, and are locally responsive to that, is shown to yield such maximal weak resilience. These results imply that locality constraints on the information available to the routing policies do not cause loss of weak resilience. Fundamental properties of dynamical networks driven by locally responsive distributed routing policies are analyzed in detail, including global convergence to a unique limit flow. The derivation of these properties exploits the cooperative nature of these dynamical systems, together with an additional stability property implied by the assumption of monotonicity of the flow as a function of the density on each link.

Linear Convergence Rate of a Class of Distributed Augmented Lagrangian Algorithms

Abstract: We study distributed optimization where nodes cooperatively minimize the sum of their individual, locally known, convex costs $f_i(x)$'s, $x \in {\mathbb R}^d$ 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.

Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes

Abstract: Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that L 1 spaces are not natural for such equations, since we lose uniqueness of the solution.

Second-Order Consensus Seeking in Multi-Agent Systems With Nonlinear Dynamics Over Random Switching Directed Networks

Abstract: This paper discusses the second-order local consensus problem for multi-agent systems with nonlinear dynamics over dynamically switching random directed networks. By applying the orthogonal decomposition method, the state vector of resulted error dynamical system can be decomposed as two transversal components, one of which evolves along the consensus manifold and the other evolves transversally with the consensus manifold. Several sufficient conditions for reaching almost surely second-order local consensus are derived for the cases of time-delay-free coupling and time-delay coupling, respectively. For the case of time-delay-free coupling, we find that if there exists one directed spanning tree in the network which corresponds to the fixed time-averaged topology and the switching rate of the dynamic network is not more than a critical value which is also estimated analytically, then second-order dynamical consensus can be guaranteed for the choice of suitable parameters. For the case of time-delay coupling, we not only prove that under some assumptions, the second-order consensus can be reached exponentially, but also give an analytical estimation of the upper bounds of convergence rate and the switching rate. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained theoretical results.

Unified Framework for Deriving Simultaneous Equation Algorithms for Water Distribution Networks

Abstract: The known formulations for steady-state hydraulics within looped water distribution networks are rederived in terms of linear and nonlinear transformations of the original set of partly linear and partly nonlinear equations that express conservation of mass and energy. All of these formulations lead to a system of nonlinear equations that can be linearized as a function of the chosen unknowns using either the Newton-Raphson (NR) or the linear theory (LT) approaches. This produces a number of different algorithms, some of which are already known in the literature, whereas others have been originally developed within this work. For the sake of clarity, all the different algorithms were rederived using the same analytical approach and a unified notation. They were all applied to the same test case network with randomly perturbed demands to compare their convergence characteristics. The results show that all of the linearly transformed formulations have exactly the same convergence rate, whose value d...

Multi-Step Gradient Methods for Networked Optimization

Abstract: We develop multi-step gradient methods for network-constrained optimization of strongly convex functions with Lipschitz-continuous gradients. Given the topology of the underlying network and bounds on the Hessian of the objective function, we determine the algorithm parameters that guarantee the fastest convergence and characterize situations when significant speed-ups over the standard gradient method are obtained. Furthermore, we quantify how uncertainty in problem data at design-time affects the run-time performance of the gradient method and its multi-step counterpart, and conclude that in most cases the multi-step method outperforms gradient descent. Finally, we apply the proposed technique to three engineering problems: resource allocation under network-wide budget constraint, distributed averaging, and Internet congestion control. In all cases, our proposed algorithms converge significantly faster than the state-of-the art.

Local Linear Convergence of the Alternating Direction Method of Multipliers for Quadratic Programs

Abstract: The Douglas--Rachford alternating direction method of multipliers (ADMM) has been widely used in various areas. The global convergence of ADMM is well known, while research on its convergence rate ...

On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations

Abstract: We study the convergence of monotone $P1$ finite element methods on unstructured meshes for fully nonlinear Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under nondegeneracy assumptions, strong $L^2$ convergence of the gradients.