DRUM :: Browsing by Author "Baras, John"

Browsing by Author "Baras, John"

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The asymptotic consensus problem on convex metric spaces
(2010-03) Matei, Ion; Baras, John; Baras, John
A consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. The agents exchange information according to a communication network modeled as a directed time-varying graph. A convex metric space is a metric space on which we define a convex structure. Using this convex structure we define convex sets and in particular the convex hull of a (finite) set. In this paper we generalize the asymptotic consensus problem to convex metric spaces. Under minimal connectivity assumptions, we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the convex hull of the agent's current state and the ones of his neighbors, the asymptotic agreement is achieved. As application example, we use this framework to introduce an iterative algorithm for reaching consensus of opinion. In this example, the agents take values in the space of discrete random variable on which we define an appropriate metric and convex structure. In addition, we provide a more detail analysis of the convex hull of a finite set for this particular convex metric space.
Compositional Analysis of Dynamic Bayesian Networks and Applications to CPS
(2012) Yang, ShahAn; Zhou, Yuchen; Baras, John
Dynamic Bayesian networks (DBNs) can be effectively used to model various problems in complex dynamic systems. We perform an empirical investigation on compositional analysis of DBNs using abstraction. In static systems and hidden Markov models, computation of a metric called treewidth induces a tree decomposition that can be used to perform logical or probabilistic inference and max+ optimizations in time exponential in treewidth and linear in overall system size. Intuitively, the linear scaling means that very large systems can be analyzed as long as they are sufficiently sparse and well structured. In these simple cases, summary propagation, which uses two operations, summation (projection) and product (composition), suffices to perform the inference or optimization. Here, we begin an extension of this to structured networks of communicating dynamic systems. We define generalizations of projection and composition operators that treat labeled Markov chains as primitive objects. The projection operation, corresponding to summation, is implemented as label deletion followed by exact state reduction for Markov chains, similar to Hopcroft’s DFA minimization algorithm, with O(n log m) complexity. The composition operation is the product of state machines. We use canonical MDDs, similar to BDDs, to capture logical dependencies symbolically. Combining symbolic representations with Markov chain lumping algorithms is a novel contribution. Using this approach, we have created a tool leveraging model based systems engineering technologies. The communicating Markov chains are specified using UML Statecharts via Papyrus extended using an ANTLR parsed domain specific language (DSL). The tool reduces the number of states in networks of Markov chains by several orders of magnitude. In one example, a network having a product state space of more than 600 million states is reduced to about 500 states. A feature of this technique is that the state space is examined incrementally, meaning that the full state space is never explicitly represented, even as an input to the reduction algorithm. The primary reduction appears to come from symmetry which is surprising because the technique includes no explicit symmetry handling. We note that composition is efficient at least for systems with high symmetry. We describe applications to a hospital intensive care unit (ICU) from a systems engineering perspective.
Consensus-Based Distributed Filtering
(2010-03) Matei, Ion; Baras, John; Baras, John
We address the consensus-based distributed linear filtering problem, where a discrete time, linear stochastic process is observed by a network of sensors. We assume that the consensus weights are known and we first provide sufficient conditions under which the stochastic process is detectable, i.e. for a specific choice of consensus weights there exists a set of filtering gains such that the dynamics of the estimation errors (without noise) is asymptotically stable. Next, we provide a distributed sub-optimal filtering scheme based on optimizing an upper bound on a quadratic filtering cost. In the stationary case, we provide sufficient conditions under which this scheme converges; conditions expressed in terms of the convergence properties of a set of coupled Riccati equations. We continue with presenting a connection between the consensus-based distributed linear filter and the optimal linear filter of a Markovian jump linear system, appropriately defined. More specifically, we show that if the Markovian jump linear system is (mean square) detectable, then the stochastic process is detectable under the consensus-based distributed linear filtering scheme. We also show that the state estimate given by the optimal linear filter of a Markovian jump linear system appropriately defined can be seen as an approximation of the optimal average state estimate obtained using the consensus-based linear filtering scheme.
Convergence Results for Ant Routing Algorithms via Stochastic Approximations
(2009-10) Purkayastha, Punyaslok; Baras, John; Baras, John
In this paper, we provide convergence results for an Ant-Based Routing (ARA) Algorithm for wireline, packet-switched communication networks, that are acyclic. Such algorithms are inspired by the foraging behavior of ants in nature. We consider an ARA algorithm proposed by Bean and Costa. The algorithm has the virtues of being adaptive and distributed, and can provide a multipath routing solution. We consider a scenario where there are multiple incoming data traffic streams that are to be routed to their respective destinations via the network. Ant packets, which are nothing but probe packets, are introduced to estimate the path delays in the network. The node routing tables, which consist of routing probabilities for the outgoing links, are updated based on these delay estimates. In contrast to the available analytical studies in the literature, the link delays in our model are stochastic, time-varying, and dependent on the link traffic. The evolution of the delay estimates and the routing probabilities are described by a set of stochastic iterative equations. In doing so, we take into account the distributed and asynchronous nature of the algorithm operation. Using methods from the theory of stochastic approximations, we show that the evolution of the delay estimates can be closely tracked by a deterministic ODE (Ordinary Differential Equation) system, when the step-size of the delay estimation scheme is small. We study the equilibrium behavior of the ODE system in order to obtain the equilibrium behavior of the routing algorithm. We also explore properties of the equilibrium routing probabilities, and provide illustrative simulation results.
Distributed algorithms for optimization problems with equality constraints
(2013-02-28) Matei, Ion; Baras, John
In this paper we introduce two discrete-time, distributed optimization algorithms executed by a set of agents whose interactions are subject to a communication graph. The algorithms can be applied to optimization problems where the cost function is expressed as a sum of functions, and where each function is associated to an agent. In addition, the agents can have equality constraints as well. The algorithms are not consensus-based and can be applied to non-convex optimization problems with equality constraints. We demonstrate that the first distributed algorithm results naturally from applying a first order method to solve the first order necessary conditions of a lifted optimization problem with equality constraints; optimization problem whose solution embeds the solution of our original problem. We show that, provided the agents’ initial values are sufficiently close to a local minimizer, and the step-size is sufficiently small, under standard conditions on the cost and constraint functions, each agent converges to the local minimizer at a linear rate. Next, we use an augmented Lagrangian idea to derive a second distributed algorithm whose local convergence requires weaker sufficient conditions than in the case of the first algorithm.
A Distributed Learning Algorithm with Bit-valued Communications for Multi-agent Welfare Optimization
(2013) Menon, Anup; Baras, John
A multi-agent system comprising N agents, each picking actions from a finite set and receiving a payoff that depends on the action of the whole, is considered. The exact form of the payoffs are unknown and only their values can be measured by the respective agents. A decentralized algorithm was proposed by Marden et. al. [1] and in the authors’ earlier work [2] that, in this setting, leads to the agents picking welfare optimizing actions under some restrictive assumptions on the payoff structure. This algorithm is modified in this paper to incorporate exchange of certain bit-valued information between the agents over a directed communication graph. The notion of an interaction graph is then introduced to encode known interaction in the system. Restrictions on the payoff structure are eliminated and conditions that guarantee convergence to welfare minimizing actions w.p. 1 are derived under the assumption that the union of the interaction graph and communication graph is strongly connected.
Distributed Opportunistic Scheduling for Wireless Ad-Hoc Networks with Block-Fading Model
(2013) Chen, Hua; Baras, John
In this paper, we study a distributed opportunistic scheduling problem to exploit the channel fluctuations in wireless ad-hoc networks. In this problem, channel probing is followed by a transmission scheduling procedure executed independently within each link in the network. We study this problem for the popular block-fading channel model, where channel dependencies are inevitable between different time instances during the channel probing phase. Different from existing works, we explicitly consider this type of channel dependencies and its impact on the transmission scheduling and hence the system performance. We use optimal stopping theory to formulate this problem, but at carefully chosen time instances at which effective decisions are made. The problem can then be solved by a new stopping rule problem where the observations are independent between different time instances. Since the stopping rule problem has an implicit horizon determined by the network size, we first characterize the system performance using backward induction. We develop one recursive approach to solve the problem and show that the computational complexity is linear with respect to network size. Due to its computational complexity, we present an approximated approach for performance analysis and develop a metric to check how good the approximation is. We characterize the achievable system performance if we ignore the finite horizon constraint and design stopping rules based on the infinite horizon analysis nevertheless. We present an improved protocol to reduce the probing costs which requires no additional cost. We characterize the performance improvement and the energy savings in terms of the probing signals. We show numerical results based on our mathematical analysis with various settings of parameters.
Distributed subgradient method under random communication topology - the e
(2009-09) Matei, Ion; Baras, John; Baras, John
In this note we study the performance metrics (rate of convergence and guaranteed region of convergence) of a multi-agent subgradient method for optimizing a sum of convex functions. We assume that the agents exchange information according to a communication topology modeled as a random graph, independent of other time instances. Under a strong convexity type of assumption, we express the performance metrics directly as functions of the estimates of the optimal decision vector. We emphasize how the probability distribution of the random graph affects the upper bounds on the performance metrics. This provide a guide for tuning the parameters of the communication protocol such that good performance of the multi-agent subgradient method is ensured. We perform the tuning of the protocol parameters for two communication scenarios. In the first scenario, we assume a randomized scheme for link activation with no-error transmissions while in the second scenario we use a pre-established order of transmissions but we consider the interference effect. Both these scenarios are applied on a small world type of topology.
Distributed Topology Control for Stable Path Routing in Mobile Ad Hoc Networks
(2009-11-25) Somasundaram, Kiran; Jain, Kaustubh; Tabatabaee, Vahid; Baras, John
Distributed Topology Control for Stable Path Routing in Multi-hop Wireless Networks
(2010-03) Somasundaram, Kiran; Baras, John; Jain, Kaustubh; Tabatabaee, Vahid
In this paper, we introduce the stable path topology control problem for routing in mobile multi-hop networks. We formulate the topology control problem of selective link-state broadcast as a graph pruning problem with restricted local neighborhood information. We develop a multi-agent optimiza- tion framework where the decision policies of each agent are restricted to local policies on incident edges and independent of the policies of the other agents. We show that under a condition called the positivity condition, these independent local policies preserve the stable routing paths globally. We then provide an efficient algorithm to compute an optimal local policy that yields a minimal pruned graph, which we call the Stable Path Topology Control (SPTC) algorithm. Using simulations, we demonstrate that this algorithm, when used with the popular ETX metric, outperforms topology control mechanisms commonly used for Mobile Ad Hoc Networks.
A Fixed Point Theory Approach to Multi-Agent Consensus Dynamics With Delays
(2013-01-01) Somarakis, Christoforos; Baras, John
The classic linear time-invariant multi-agent consensus scheme is revisited in the presence of constant and distributed bounded delays. We create a fixed point argument and prove exponential convergence with specific rate that depends both on the topology of the communication graph and the upper bound of the allowed delay.
Flow Control in Time-Varying, Random Supply Chains
(2012-02) Matei, Ion; Gueye, Assane; Baras, John
Today’s supply chains are more and more complex. They depend on a network of independent, yet interconnected moving parts. They rely on critical infrastructures and experience a lot of time variability and randomness. Designing strategies that deal with such constantly changing supply chains is necessary in this increasingly globalized economy where supply chain disruptions have impacts that propagate not only locally but also globally. In this paper we propose a randomized flow control algorithm for a time varying, random supply chain network. We formulate a constrained stochastic optimization problem that maximizes the profit function in terms of the long-run, time-average rates of the flows in the supply chain. We show that our algorithm, which is based on queueing theory and stochastic analysis concepts, can get arbitrarily close to the solution of the aforementioned optimization problem. In addition, we describe how the flow control algorithm can be extended to a multiple firms supply chain setup and present numerical simulations of our algorithm for different supply chain topologies.
A non-consensus based distributed optimization algorithm
(2013-02-27) Matei, Ion; Baras, John
In this paper we introduce a discrete-time, distributed optimization algorithm executed by a set of agents whose interactions are subject to a communication graph. The algorithm can be applied to optimization costs that are expressed as sums of functions, where each function is associated to an agent. The algorithm can be applied to continuously differentiable cost functions, it is not consensus-based and is derived naturally by solving the first order necessary conditions of a lifted optimization problem with equality constraints. We show that, provided the agents’ initial values are sufficiently closed to a local minimizer and the step-size is sufficiently small, each agent converges to the local minimizer at a linear rate. In addition, we revisit two popular consensus-based distributed optimization algorithms and give sufficient conditions so that there use is extended to non-convex functions as well. We take a closer look at their rate of convergence and also show that unlike our algorithm, for a constant step-size, the consensus-based algorithms do not converge to a local minimizer even though the agents start close enough to the local minimizer.
A non-heuristic distributed algorithm for non-convex constrained optimization
(2013-02-15) Matei, Ion; Baras, John
In this paper we introduce a discrete-time, distributed optimization algorithm executed by a set of agents whose interactions are subject to a communication graph. The algorithm can be applied to optimization problems where the cost function is expressed as a sum of functions, and where each function is associated to an agent. In addition, the agents can have equality constraints as well. The algorithm can be applied to non-convex optimization problems with equality constraints, it is not consensus-based and it is not an heuristic. We demonstrate that the distributed algorithm results naturally from applying a first order method to solve the first order necessary conditions of an augmented optimization problem with equality constraints; optimization problem whose solution embeds the solution of our original problem. We show that, provided the agents’ initial values are sufficiently close to a local minimum, and the step-size is sufficiently small, under standard conditions on the cost and constraint functions, each agent converges to the local minimum at a linear rate.
Nonlinear Programming Methods for Distributed Optimization
(2015-01) Matei, Ion; Baras, John
In this paper we investigate how standard nonlinear programming algorithms can be used to solve constrained optimization problems in a distributed manner. The optimization setup consists of a set of agents interacting through a communication graph that have as common goal the minimization of a function expressed as a sum of (possibly non-convex) differentiable functions. Each function in the sum corresponds to an agent and each agent has associated an equality constraint. By re-casting the distributed optimization problem into an equivalent, augmented centralized problem, we show that distributed algorithms result naturally from applying standard nonlinear programming tech- niques. Due to the distributed formulation, the standard assumptions and convergence results no longer hold. We emphasize what changes are necessary for convergence to still be achieved for three algorithms: two algorithms based on Lagrangian methods, and an algorithm based the method of multipliers. The changes in the convergence results are necessary mainly due to the fact that the (local) minimizers of the lifted optimization problem are not regular, as a results of the distributed formulation. Unlike the standard algorithm based on the method of multipliers, for the distributed version we cannot show that the theoretical super-linear convergence rate can be achieved.
On the dynamics of linear functional differential equations with asymptotically constant solutions
(2014-08-22) Somarakis, Christoforos; Baras, John; Paraskevas, Evripidis; Baras, John
We discuss the dynamics of general linear functional differential equations with solutions that exhibit asymptotic constancy. We apply fixed point theory methods to study the stability of these solutions and we provide sufficient conditions of asymptotic stability with emphasis on the rate of convergence. Several examples are provided to illustrate the claim that the derived results generalize, unify and in some cases improve the existing ones.
Pareto Nash Replies for Multi-Objective Games
(2008) Somasundaram, Kiran; Baras, John; Baras, John
Path Optimization Techniques for Trusted Routing in Mobile Ad-Hoc Networks: An Interplay Between Ordered Semirings
(2008) Somasundaram, Kiran; Baras, John; Baras, John
In this paper, we formulate the problem of trusted routing as a transaction of services over a complex networked environment. We present definitions from service-oriented environments which unambiguously capture the difference between trust and reputation relations. We show that the trustworthiness measures associated with these relations have a linear order embedded in them. Identifying this order structure permits us to treat the trusted routing problem as a bi-objective path optimization problem. Further, we present polynomial time solutions to obtain the optimal routing paths in various biobjective settings. In developing these algorithms, we identify an interesting semiring decomposition principle that yields a distributed solution
Path Optimization Techniques for Trusted Routing in Tactical Mobile Ad-Hoc Networks
(2008) Somasundaram, Kiran; Baras, John; Baras, John
A performance comparison between two consensus-based distributed optimization algorithms
(2012-05-04) Matei, Ion; Baras, John
In this paper we address the problem of multi-agent optimization for convex functions expressible as sums of convex functions. Each agent has access to only one function in the sum and can use only local information to update its current estimate of the optimal solution. We consider two consensus-based iterative algorithms, based on a combination between a consensus step and a subgradient decent update. The main difference between the two algorithms is the order in which the consensus-step and the subgradient descent update are performed. We show that updating first the current estimate in the direction of a subgradient and then executing the consensus step ensures better performance than executing the steps in reversed order. In support of our analytical results, we give some numerical simulations of the algorithms as well.

Browsing by Author "Baras, John"

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