approximate primal solutions and rate analysis for dual subgradient methods in this paper we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the lagrangian dual of a primal convex possibly nondifferentiable constrained optimization problem our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods such as the recent resource allocation problems in large scale networks for such problems we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions these algorithms use a constant stepsize in view of its simplicity and practical significance we provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions these estimates are given per iteration thus providing a basis for analyzing the trade offs between the desired level of error and the selection of the stepsize value our analysis relies on the slater condition and the inherited boundedness properties of the dual problem under this condition it also relies on the boundedness of subgradients which is ensured by assuming the compactness of the constraint set

determination of the algebraic relations among special γ values in positive characteristic we devise a new criterion for linear independence over function fields using this tool in the setting of dual t motives we find that all algebraic relations among special values of the geometric r function over f q t are explained by the standard functional equations

on the optimal parameter of the composite laplacian quadratics function abstract recently the composite quadratic lyapunov function has been extended to study of multi agent systems leading to the so called composite laplacian quadratics clqs function compared with quadratic lyapunov functions the clqs function can yield a larger convergence region and is particularly useful in stabilization of multi agent systems with complex dynamics such as differential inclusions in the definition of the clqs function an optimal vector parameter plays a critical role in determining the value of the clqs function and in constructing stabilization laws derived from the clqs function this paper focuses on the properties of the optimal parameter of the clqs function the uniqueness of the optimal parameter is established a distributed computation approach is further proposed which is useful in computing the optimal parameter the robustness issue of the optimal parameter is also investigated for a multi agent system described by linear differential inclusions finally a numerical example is provided to validate the proposed theoretical results

distributed continuous time approximate projection protocols for shortest distance optimization problems in this paper we investigate a distributed shortest distance optimization problem for a multi agent network to cooperatively minimize the sum of the quadratic distances from some convex sets where each set is only associated with one agent to deal with this optimization problem with projection uncertainties we propose a distributed continuous time dynamical protocol where each agent can only obtain an approximate projection and communicate with its neighbors over a time varying communication graph first we show that no matter how large the approximate angle is system states are always bounded for any initial condition and uniformly bounded with respect to all initial conditions if the inferior limit of the stepsize is greater than zero then in both cases of nonempty and empty intersection of convex sets we provide stepsize and approximate angle conditions to ensure the optimal convergence respectively moreover we also give some characterizations about the optimal solutions for the empty intersection case

on distributed convex optimization under inequality and equality constraints we consider a general multi agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint a global equality constraint and a global constraint set the objective function is defined by a sum of local objective functions while the global constraint set is produced by the intersection of local constraint sets in particular we study two cases one where the equality constraint is absent and the other where the local constraint sets are identical we devise two distributed primal dual subgradient algorithms based on the characterization of the primal dual optimal solutions as the saddle points of the lagrangian and penalty functions these algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the slater s condition

distributed algorithms for the computation of noncooperative equilibria abstract in this paper a general class of nonquadratic convex nash games is studied from the points of view of existence stability and iterative computation of noncooperative equilibria conditions for contraction of general nonlinear operators are obtained which are then used in the stability study of such games these lead to existence and uniqueness conditions for stable nash equilibrium solutions under both global and local analysis also convergence of an algorithm which employs inaccurate search techniques is verified it is shown in the context of a fish war example that the algorithm given is in some aspects superior to various algorithms found in the literature and is furthermore more meaningful for real world implementation

distributed convergence to nash equilibria in two network zero sum games this paper considers a class of strategic scenarios in which two networks of agents have opposing objectives with regard to the optimization of a common objective function in the resulting zero sum game individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network for the case when the interaction topology of each network is undirected we synthesize a distributed saddle point strategy and establish its convergence to the nash equilibrium for the class of strictly concave convex and locally lipschitz objective functions we also show that this dynamics does not converge in general if the topologies are directed this justifies the introduction in the directed case of a generalization of this distributed dynamics which we show converges to the nash equilibrium for the class of strictly concave convex differentiable functions with globally lipschitz gradients the technical approach combines tools from algebraic graph theory nonsmooth analysis set valued dynamical systems and game theory

inexact dual averaging method for distributed multi agent optimization abstract we consider a distributed convex optimization problem over a network where multiple agents collectively try to minimize a sum of local convex functions of the same variables each of which is available to one specific agent only for solving this optimization problem we present an inexact version of the dual averaging method this extends recent results of duchi 2012 which cover the error free case to the case where an error is present in calculating the subgradient of the objective function or in computing the projection we show that when the errors decrease at appropriate rates our method achieves the same convergence rate as in the error free case in particular the convergence of the method is also established for nonsummable errors we also provide numerical results to validate the theoretic results

distributed seeking of nash equilibria with applications to mobile sensor networks we consider the problem of distributed convergence to a nash equilibrium in a noncooperative game where the players generate their actions based only on online measurements of their individual cost functions corrupted with additive measurement noise exact analytical forms and or parameters of the cost functions as well as the current actions of the players may be unknown additionally the players actions are subject to linear dynamic constraints we propose an algorithm based on discrete time stochastic extremum seeking using sinusoidal perturbations and prove its almost sure convergence to a nash equilibrium we show how the proposed algorithm can be applied to solving coordination problems in mobile sensor networks where motion dynamics of the players can be modeled as 1 single integrators velocity actuated vehicles 2 double integrators force actuated vehicles and 3 unicycles a kinematic model with nonholonomic constraints examples are given in which the cost functions are selected such that the problems of connectivity control formation control rendezvous and coverage control are solved in an adaptive and distributed way the methodology is illustrated through simulations

distributed averaging on digital erasure networks iterative distributed algorithms are studied for computing arithmetic averages over networks of agents connected through memoryless broadcast erasure channels these algorithms do not require the agents to have any knowledge about the global network structure or size almost sure convergence to state agreement is proved and the communication and computational complexities of the algorithms are analyzed both the number of transmissions and the number of computations performed by each agent of the network are shown to grow not faster than poly logarithmically in the desired precision the impact of the graph topology on the algorithms performance is analyzed as well moreover it is shown how in the presence of noiseless communication feedback one can modify the algorithms significantly improving their performance versus complexity trade off

constrained consensus and optimization in multi agent networks we present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time varying connectivity our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem where the global objective function is a combination of local agent objective functions our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets to highlight the effects of constraints we first consider a constrained consensus problem and present a distributed projected consensus algorithm in which agents combine their local averaging operation with projection on their individual constraint sets this algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents we establish convergence and convergence rate results for the projected consensus algorithm we next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets we present a distributed projected subgradient algorithm which involves each agent performing a local averaging operation taking a subgradient step to minimize its own objective function and projecting on its constraint set we show that with an appropriately selected stepsize rule the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal and when the weights are time varying but all agents have the same constraint set

stability of primal dual gradient dynamics and applications to network optimization this paper considers dynamic laws that seek a saddle point of a function of two vector variables by moving each in the direction of the corresponding partial gradient this method has old roots in the classical work of arrow hurwicz and uzawa on convex optimization and has seen renewed interest with its recent application to resource allocation in communication networks this paper brings other tools to bear on this problem in particular krasovskii s method to find lyapunov functions and recently obtained extensions of the lasalle invariance principle for hybrid systems these methods are used to obtain stability proofs of these primal dual laws in different scenarios and applications to cross layer network optimization are exhibited

a randomized incremental subgradient method for distributed optimization in networked systems we present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to nedic and bertsekas siam j optim 12 2001 pp 109 138 our novel algorithm is particularly suitable for distributed implementation and execution and possible applications include distributed optimization e g parameter estimation in networks of tiny wireless sensors the stochastic component in the algorithm is described by a markov chain which can be constructed in a distributed fashion using only local information we provide a detailed convergence analysis of the proposed algorithm and compare it with existing both deterministic and randomized incremental subgradient methods

on entropy consistency of large time step schemes i the godunov and glimm schemes this paper shows for scalar conservation laws with a convex flux function that the large time step approximation of weak solutions gives entropy solutions in the limit if the courant number is between tfrac 1 2 and 1 for flux functions with constant curvature monotonicity or monotone initial data the results extend to larger courant numbers a continuity argument gives this result also for the case of fluxes with almost constant curvature

a necessary and sufficient condition for consensus over random networks we consider the consensus problem for stochastic discrete time linear dynamical systems the underlying graph of such systems at a given time instance is derived from a random graph process independent of other time instances for such a framework we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments this easily verifiable condition uses the spectrum of the average weight matrix finally we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1

dual decomposition for multi agent distributed optimization with coupling constraints abstract we study distributed optimization in a cooperative multi agent setting where agents have to agree on the usage of shared resources and can communicate via a time varying network to this purpose each agent has its own decision variables that should be set so as to minimize its individual objective function subject to local constraints resource sharing is modeled via coupling constraints that involve the non positivity of the sum of agents individual functions each one depending on the decision variables of one single agent we propose a novel distributed algorithm to minimize the sum of the agents objective functions subject to both local and coupling constraints where dual decomposition and proximal minimization are combined in an iterative scheme notably privacy of information is guaranteed since only the dual optimization variables associated with the coupling constraints are exchanged by the agents under convexity assumptions jointly with suitable connectivity properties of the communication network we are able to prove that agents reach consensus to some optimal solution of the centralized dual problem counterpart while primal variables converge to the set of optimizers of the centralized primal problem the efficacy of the proposed approach is demonstrated on a plug in electric vehicles charging problem

convergence rates of distributed nesterov like gradient methods on random networks we consider distributed optimization in random networks where n nodes cooperatively minimize the sum i 1n fi x of their individual convex costs existing literature proposes distributed gradient like methods that are computationally cheap and resilient to link failures but have slow convergence rates in this paper we propose accelerated distributed gradient methods that 1 are resilient to link failures 2 computationally cheap and 3 improve convergence rates over other gradient methods we model the network by a sequence of independent identically distributed random matrices w k drawn from the set of symmetric stochastic matrices with positive diagonals the network is connected on average and the cost functions are convex differentiable with lipschitz continuous and bounded gradients we design two distributed nesterov like gradient methods that modify the d ng and d nc methods that we proposed for static networks we prove their convergence rates in terms of the expected optimality gap at the cost function let k and k be the number of per node gradient evaluations and per node communications respectively then the modified d ng achieves rates o logk k and o logk k and the modified d nc rates o 1 k2 and o 1 k2 where 0 is arbitrarily small for comparison the standard distributed gradient method cannot do better than 1 k2 3 and 1 k2 3 on the same class of cost functions even for static networks simulation examples illustrate our analytical findings

a minimax theorem with applications to machine learning signal processing and finance this paper concerns a fractional function of the form x ta sqrt x tbx where b is positive definite we consider the game of choosing x from a convex set to maximize the function and choosing a b from a convex set to minimize it we prove the existence of a saddle point and describe an efficient method based on convex optimization for computing it we describe applications in machine learning robust fisher linear discriminant analysis signal processing robust beamforming and robust matched filtering and finance robust portfolio selection in these applications x corresponds to some design variables to be chosen and the pair a b corresponds to the statistical model which is uncertain

a class of large time step godunov schemes for hyperbolic conservation laws and applications a large time step lts godunov scheme firstly proposed by leveque is further developed in the present work and applied to euler equations based on the analysis of the computational performances of leveque s linear approximation on wave interactions a multi wave approximation on rarefaction fan is proposed to avoid the occurrences of rarefaction shocks in computations the developed lts scheme is validated using 1 d test cases manifesting high resolution for discontinuities and the capability of maintaining computational stability when large cfl numbers are imposed the scheme is then extended to multidimensional problems using dimensional splitting technique the treatment of boundary condition for this multidimensional lts scheme is also proposed as for demonstration problems inviscid flows over naca0012 airfoil and onera m6 wing with given swept angle are simulated using the developed lts scheme the numerical results reveal the high resolution nature of the scheme where the shock can be captured within 1 2 grid points the resolution of the scheme would improve gradually along with the increasing of cfl number under an upper bound where the solution becomes severely oscillating across the shock computational efficiency comparisons show that the developed scheme is capable of reducing the computational time effectively with increasing the time step cfl number

the role of convexity in saddle point dynamics lyapunov function and robustness this paper studies the projected saddle point dynamics associated to a convex concave function which we term saddle function the dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and projected gradient ascent in variables corresponding to concavity we examine the role that the local and or global nature of the convexity concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics under the assumption that the saddle function is twice continuously differentiable we provide a novel characterization of the omega limit set of the trajectories of this dynamics in terms of the diagonal blocks of the hessian using this characterization we establish global asymptotic convergence of the dynamics under local strong convexity concavity of the saddle function when strong convexity concavity holds globally we establish three results first we identify a lyapunov function that decreases strictly along the trajectory for the projected saddle point dynamics when the saddle function corresponds to the lagrangian of a general constrained convex optimization problem second for the particular case when the saddle function is the lagrangian of an equality constrained optimization problem we show input to state stability iss of the saddle point dynamics by providing an iss lyapunov function third we use the latter result to design an opportunistic state triggered implementation of the dynamics various examples illustrate our results

convergence speed in distributed consensus and averaging we study the convergence speed of distributed iterative algorithms for the consensus and averaging problems with emphasis on the latter we first consider the case of a fixed communication topology we show that a simple adaptation of a consensus algorithm leads to an averaging algorithm we prove lower bounds on the worst case convergence time for various classes of linear time invariant distributed consensus methods and provide an algorithm that essentially matches those lower bounds we then consider the case of a time varying topology and provide a polynomial time averaging algorithm

distributed nash equilibrium seeking this paper presents an asynchronous gossip based algorithm for finding a nash equilibrium ne of a game in a distributed multi player network the algorithm is designed in such a way that players make decisions based on estimates of the other players actions obtained from local neighbors using a set of standard assumptions on the cost functions and communication graph the paper proves almost sure convergence to a ne for diminishing step sizes for constant step sizes an error bound on expected distance from a ne is established the effectiveness of the proposed algorithm is demonstrated via simulation for both diminishing and constant step sizes

nash equilibrium computation in subnetwork zero sum games with switching communications in this paper we investigate a distributed nash equilibrium computation problem for a time varying multi agent network consisting of two subnetworks where the two subnetworks share the same objective function we first propose a subgradient based distributed algorithm with heterogeneous stepsizes to compute a nash equilibrium of a zero sum game we then prove that the proposed algorithm can achieve a nash equilibrium under uniformly jointly strongly connected ujsc weight balanced digraphs with homogenous stepsizes moreover we demonstrate that for weighted unbalanced graphs a nash equilibrium may not be achieved with homogenous stepsizes unless certain conditions on the objective function hold we show that there always exist heterogeneous stepsizes for the proposed algorithm to guarantee that a nash equilibrium can be achieved for ujsc digraphs finally in two standard weight unbalanced cases we verify the convergence to a nash equilibrium by adaptively updating the stepsizes along with the arc weights in the proposed algorithm

extensions of fast lipschitz optimization the need of fast distributed solvers for optimization problems in networked systems has motivated the recent development of the fast lipschitz optimization framework in such an optimization problems satisfying certain qualifying conditions such as monotonicity of the objective function and contractivity of the constraints have a unique optimal solution obtained via fast distributed algorithms that compute the fixed point of the constraints this paper extends the set of problems for which the fast lipschitz framework applies existing assumptions on the problem form are relaxed and new and generalized qualifying conditions are established by novel results based on lagrangian duality it is shown for which cases of more constraints than decision variables and less constraints than decision variables fast lipschitz optimization applies new results are obtained by imposing non strict monotonicity of the objective functions the extended fast lipschitz framework is illustrated by a number of examples including network optimization and optimal control problems

distributed and recursive parameter estimation in parametrized linear state space models we consider a network of sensors deployed to sense a spatio temporal field and infer parameters of interest about the field we are interested in the case where each sensor s observation sequence is modeled as a state space process that is perturbed by random noise and the models across sensors are parametrized by the same parameter vector the sensors collaborate to estimate this parameter from their measurements and to this end we propose a distributed and recursive estimation algorithm which we refer to as the incremental recursive prediction error algorithm this algorithm has the distributed property of incremental gradient algorithms and the on line property of recursive prediction error algorithms

distributed algorithms for reaching consensus on general functions this paper presents analysis and design results for distributed consensus algorithms in multi agent networks we consider continuous consensus functions of the initial state of the network agents under mild smoothness assumptions we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus this characterization is the building block to obtain various design results for networks with weighted directed interconnection topologies we first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus we apply this result to the family of weighted power mean functions and characterize the exponential convergence properties of the resulting algorithms we establish the validity of these results for scenarios with switching interconnection topologies finally we conclude with two discontinuous distributed algorithms that achieve respectively max and min consensus in finite time

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

fixed time stable gradient flows applications to continuous time optimization this paper proposes novel gradient flow schemes that yield convergence to the optimal point of a convex optimization problem within a fixed time from any given initial condition for unconstrained optimization constrained optimization and min max problems it is shown that the solution of the modified gradient flow dynamics exists and is unique under certain regularity conditions on the objective function while fixed time convergence to the optimal point is shown via lyapunov based analysis the application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient dominance a relaxation of strong convexity then a modified newton s method is presented that exhibits fixed time convergence under some mild conditions on the objective function building upon this method a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed finally the general min max problem is considered and a modified scheme to obtain the optimal solution of saddle point dynamics in fixed time is developed

on entropy consistency of large time step schemes ii approximate riemann solvers it is shown that for scalar conservation laws with a convex flux function any sequence of approximate solutions constructed by large time step schemes using approximate riemann solvers as a building block gives the unique entropy solution in the limit of decreasing mesh size if the courant number is less than 1 when the curvature of the flux function is nearly constant the above results may be extended to courant numbers slightly larger than 1

nash equilibrium seeking in noncooperative games we introduce a non model based approach for locally stable convergence to nash equilibria in static noncooperative games with n players in classical game theory algorithms each player employs the knowledge of the functional form of his payoff and the knowledge of the other players actions whereas in the proposed algorithm the players need to measure only their own payoff values this strategy is based on the extremum seeking approach which has previously been developed for standard optimization problems and employs sinusoidal perturbations to estimate the gradient we consider static games with quadratic payoff functions before generalizing our results to games with non quadratic payoff functions that are the output of a dynamic system specifically we consider general nonlinear differential equations with n inputs and n outputs where in the steady state the output signals represent the payoff functions of a noncooperative game played by the steady state values of the input signals we employ the standard local averaging theory and obtain local convergence results for both quadratic payoffs where the actual convergence is semi global and non quadratic payoffs where the potential existence of multiple nash equilibria precludes semi global convergence our convergence conditions coincide with conditions that arise in model based nash equilibrium seeking however in our framework the user is not meant to check these conditions because the payoff functions are presumed to be unknown for non quadratic payoffs convergence to a nash equilibrium is not perfect but is biased in proportion to the perturbation amplitudes and the higher derivatives of the payoff functions we quantify the size of these residual biases and confirm their existence numerically in an example noncooperative game in this example we present the first application of extremum seeking with projection to ensure that the players actions remain in a given closed and bounded action set

equilibrium topology of multi agent systems with two leaders it is typical in containment control problems to assume that there is no conflict of interest among leaders in this paper we consider the situation where there is conflict between leaders namely the leaders compete to attract followers the strategies of each leader are defined by choosing at most k followers to propagate their information then we formulate a standard two player zero sum game by using graph theory and matrix theory we further prove that each player will choose exactly k followers when the game achieves a nash equilibrium it is noteworthy that the interaction graph here is generated from the conflict between leaders and then the nash equilibrium point of the game corresponds to the equilibrium topology for the case of choosing one follower a necessary and sufficient condition for an interaction graph to be the equilibrium topology is derived moreover we can obtain the equilibrium topology directly if followers interaction graph is a circulant graph or a graph with a center vertex simulation examples are provided to validate the effectiveness of the theoretical results

randomized gradient free method for multiagent optimization over time varying networks in this brief we consider the multiagent optimization over a network where multiple agents try to minimize a sum of nonsmooth but lipschitz continuous functions subject to a convex state constraint set the underlying network topology is modeled as time varying we propose a randomized derivative free method where in each update the random gradient free oracles are utilized instead of the subgradients sgs in contrast to the existing work we do not require that agents are able to compute the sgs of their objective functions we establish the convergence of the method to an approximate solution of the multiagent optimization problem within the error level depending on the smoothing parameter and the lipschitz constant of each agent s objective function finally a numerical example is provided to demonstrate the effectiveness of the method

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper