1,987 research outputs found
A Cyclic Douglas-Rachford Iteration Scheme
In this paper we present two Douglas-Rachford inspired iteration schemes
which can be applied directly to N-set convex feasibility problems in Hilbert
space. Our main results are weak convergence of the methods to a point whose
nearest point projections onto each of the N sets coincide. For affine
subspaces, convergence is in norm. Initial results from numerical experiments,
comparing our methods to the classical (product-space) Douglas-Rachford scheme,
are promising.Comment: 22 pages, 7 figures, 4 table
The Cyclic Douglas-Rachford Method for Inconsistent Feasibility Problems
We analyse the behaviour of the newly introduced cyclic Douglas-Rachford
algorithm for finding a point in the intersection of a finite number of closed
convex sets. This work considers the case in which the target intersection set
is possibly empty.Comment: 13 pages, 2 figures; references updated, figure 2 correcte
Markov decision processes and discrete-time mean-field games constrained with costly observations: Stochastic optimal control constrained with costly observations
In this thesis, we consider Markov decision processes with actively controlled observations. Optimal strategies involve the optimisation of observation times as well as the subsequent action values. We first consider an observation cost model, where the underlying state is observed only at chosen observation times at a cost. By including the time elapsed from observations as part of the augmented Markov system, the value function satisfies a system of quasi-variational inequalities (QVIs). Such a class of QVIs can be seen as an extension to the interconnected obstacle problem. We prove a comparison principle for this class of QVIs, which implies uniqueness of solutions to our proposed problem. Penalty methods are then utilised to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate this model.
We then consider a model where agents can exercise control actions that affect their speed of access to information. The agents can dynamically decide to receive observations with less delay by paying higher observation costs. Agents seek to exploit their active information gathering by making further decisions to influence their state dynamics to maximise rewards. We also extend this notion to a corresponding mean-field game (MFG). In the mean field equilibrium, each generic agent solves individually a partially observed Markov decision problem in which the way partial observations are obtained is itself also subject to dynamic control actions by the agent. Based on a finite characterisation of the agents’ belief states, we show how the mean field game with controlled costly information access can be formulated as an equivalent standard mean field game on a suitably augmented but finite state space. We prove that with sufficient entropy regularisation, a fixed point iteration converges to the unique MFG equilibrium and yields an approximate ε-Nash equilibrium for a large but finite population size. We illustrate our MFG by an example from epidemiology, where medical testing results at different speeds and costs can be chosen by the agents
Use of Devolved Controllers in Data Center Networks
In a data center network, for example, it is quite often to use controllers
to manage resources in a centralized man- ner. Centralized control, however,
imposes a scalability problem. In this paper, we investigate the use of
multiple independent controllers instead of a single omniscient controller to
manage resources. Each controller looks after a portion of the network only,
but they together cover the whole network. This therefore solves the
scalability problem. We use flow allocation as an example to see how this
approach can manage the bandwidth use in a distributed manner. The focus is on
how to assign components of a network to the controllers so that (1) each
controller only need to look after a small part of the network but (2) there is
at least one controller that can answer any request. We outline a way to
configure the controllers to fulfill these requirements as a proof that the use
of devolved controllers is possible. We also discuss several issues related to
such implementation.Comment: Appears in INFOCOM 2011 Cloud Computing Worksho
Global Behavior of the Douglas-Rachford Method for a Nonconvex Feasibility Problem
In recent times the Douglas-Rachford algorithm has been observed empirically
to solve a variety of nonconvex feasibility problems including those of a
combinatorial nature. For many of these problems current theory is not
sufficient to explain this observed success and is mainly concerned with
questions of local convergence. In this paper we analyze global behavior of the
method for finding a point in the intersection of a half-space and a
potentially non-convex set which is assumed to satisfy a well-quasi-ordering
property or a property weaker than compactness. In particular, the special case
in which the second set is finite is covered by our framework and provides a
prototypical setting for combinatorial optimization problems
Markov decision processes with observation costs: framework and computation with a penalty scheme
We consider Markov decision processes where the state of the chain is only given at chosen
observation times and of a cost. Optimal strategies involve the optimisation of observation
times as well as the subsequent action values. We consider the finite horizon and discounted
infinite horizon problems, as well as an extension with parameter uncertainty. By including
the time elapsed from observations as part of the augmented Markov system, the value function
satisfies a system of quasi-variational inequalities (QVIs). Such a class of QVIs can be seen as an
extension to the interconnected obstacle problem. We prove a comparison principle for this class
of QVIs, which implies uniqueness of solutions to our proposed problem. Penalty methods are
then utilised to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments
on three applications which illustrate our framework
Mean-field games of speedy information access with observation costs
We investigate a mean-field game (MFG) in which agents can exercise control
actions that affect their speed of access to information. The agents can
dynamically decide to receive observations with less delay by paying higher
observation costs. Agents seek to exploit their active information gathering by
making further decisions to influence their state dynamics to maximize rewards.
In the mean field equilibrium, each generic agent solves individually a
partially observed Markov decision problem in which the way partial
observations are obtained is itself also subject of dynamic control actions by
the agent. Based on a finite characterisation of the agents' belief states, we
show how the mean field game with controlled costly information access can be
formulated as an equivalent standard mean field game on a suitably augmented
but finite state space.We prove that with sufficient entropy regularisation, a
fixed point iteration converges to the unique MFG equilibrium and yields an
approximate -Nash equilibrium for a large but finite population size.
We illustrate our MFG by an example from epidemiology, where medical testing
results at different speeds and costs can be chosen by the agents.Comment: 33 pages, 4 figure
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