212 research outputs found
Guaranteed robustness properties of multivariable, nonlinear, stochastic optimal regulators
The robustness of optimal regulators for nonlinear, deterministic and stochastic, multi-input dynamical systems is studied under the assumption that all state variables can be measured. It is shown that, under mild assumptions, such nonlinear regulators have a guaranteed infinite gain margin; moreover, they have a guaranteed 50 percent gain reduction margin and a 60 degree phase margin, in each feedback channel, provided that the system is linear in the control and the penalty to the control is quadratic, thus extending the well-known properties of LQ regulators to nonlinear optimal designs. These results are also valid for infinite horizon, average cost, stochastic optimal control problems
Qualitative properties of -fair policies in bandwidth-sharing networks
We consider a flow-level model of a network operating under an -fair
bandwidth sharing policy (with ) proposed by Roberts and
Massouli\'{e} [Telecomunication Systems 15 (2000) 185-201]. This is a
probabilistic model that captures the long-term aspects of bandwidth sharing
between users or flows in a communication network. We study the transient
properties as well as the steady-state distribution of the model. In
particular, for , we obtain bounds on the maximum number of flows
in the network over a given time horizon, by means of a maximal inequality
derived from the standard Lyapunov drift condition. As a corollary, we
establish the full state space collapse property for all . For the
steady-state distribution, we obtain explicit exponential tail bounds on the
number of flows, for any , by relying on a norm-like Lyapunov
function. As a corollary, we establish the validity of the diffusion
approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009)
1719-1780], in steady state, for the case where and under a local
traffic condition.Comment: Published in at http://dx.doi.org/10.1214/12-AAP915 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Optimization of Multiclass Queueing Networks: Polyhedral and Nonlinear Characterizations of Achievable Performance
We consider open and closed multiclass queueing networks with Poisson arrivals (in open networks), exponentially distributed class dependent service times, and with class dependent deterministic or probabilistic routing. For open networks, the performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose variants of a method to derive polyhedral and nonlinear spaces which contain the entire set of achievable response times under stable and preemptive scheduling policies. By optimizing over these spaces, we obtain lower bounds on achievable performance. In particular, we obtain a sequence of progressively more complicated nonlinear approximations (relaxations) which are progressively closer to the exact achievable space. In the special case of single station networks (multiclass queues and Klimov's model) and homogenous multiclass networks, our characterization gives exactly the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. For closed networks, the performance objective is to maximize throughput. We similarly find polyhedral and nonlinear spaces that include the performance space and by maximizing over these spaces we obtain an upper bound on the optimal throughput. We check the tightness of our bounds by simulating heuristic scheduling policies for simple open networks and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with both the number of variables and constraints being polynomial (quadratic) in the number of classes in the network. The i-th order approximation involves solving a convex programming problem in dimension O(Ri+l), where R is the number of classes in the network, which can be solved efficiently using techniques from semi-definite programming
A Dynamic Programming Approach to Adaptive Fractionation
We conduct a theoretical study of various solution methods for the adaptive
fractionation problem. The two messages of this paper are: (i) dynamic
programming (DP) is a useful framework for adaptive radiation therapy,
particularly adaptive fractionation, because it allows us to assess how close
to optimal different methods are, and (ii) heuristic methods proposed in this
paper are near-optimal, and therefore, can be used to evaluate the best
possible benefit of using an adaptive fraction size.
The essence of adaptive fractionation is to increase the fraction size when
the tumor and organ-at-risk (OAR) are far apart (a "favorable" anatomy) and to
decrease the fraction size when they are close together. Given that a fixed
prescribed dose must be delivered to the tumor over the course of the
treatment, such an approach results in a lower cumulative dose to the OAR when
compared to that resulting from standard fractionation. We first establish a
benchmark by using the DP algorithm to solve the problem exactly. In this case,
we characterize the structure of an optimal policy, which provides guidance for
our choice of heuristics. We develop two intuitive, numerically near-optimal
heuristic policies, which could be used for more complex, high-dimensional
problems. Furthermore, one of the heuristics requires only a statistic of the
motion probability distribution, making it a reasonable method for use in a
realistic setting. Numerically, we find that the amount of decrease in dose to
the OAR can vary significantly (5 - 85%) depending on the amount of motion in
the anatomy, the number of fractions, and the range of fraction sizes allowed.
In general, the decrease in dose to the OAR is more pronounced when: (i) we
have a high probability of large tumor-OAR distances, (ii) we use many
fractions (as in a hyper-fractionated setting), and (iii) we allow large daily
fraction size deviations.Comment: 17 pages, 4 figures, 1 tabl
Evolutionary game of coalition building under external pressure
We study the fragmentation-coagulation (or merging and splitting)
evolutionary control model as introduced recently by one of the authors, where
small players can form coalitions to resist to the pressure exerted by the
principal. It is a Markov chain in continuous time and the players have a
common reward to optimize. We study the behavior as grows and show that the
problem converges to a (one player) deterministic optimization problem in
continuous time, in the infinite dimensional state space
Exploring Graphs with Time Constraints by Unreliable Collections of Mobile Robots
A graph environment must be explored by a collection of mobile robots. Some
of the robots, a priori unknown, may turn out to be unreliable. The graph is
weighted and each node is assigned a deadline. The exploration is successful if
each node of the graph is visited before its deadline by a reliable robot. The
edge weight corresponds to the time needed by a robot to traverse the edge.
Given the number of robots which may crash, is it possible to design an
algorithm, which will always guarantee the exploration, independently of the
choice of the subset of unreliable robots by the adversary? We find the optimal
time, during which the graph may be explored. Our approach permits to find the
maximal number of robots, which may turn out to be unreliable, and the graph is
still guaranteed to be explored.
We concentrate on line graphs and rings, for which we give positive results.
We start with the case of the collections involving only reliable robots. We
give algorithms finding optimal times needed for exploration when the robots
are assigned to fixed initial positions as well as when such starting positions
may be determined by the algorithm. We extend our consideration to the case
when some number of robots may be unreliable. Our most surprising result is
that solving the line exploration problem with robots at given positions, which
may involve crash-faulty ones, is NP-hard. The same problem has polynomial
solutions for a ring and for the case when the initial robots' positions on the
line are arbitrary.
The exploration problem is shown to be NP-hard for star graphs, even when the
team consists of only two reliable robots
Parameterized Supply Function Bidding: Equilibrium and Efficiency
We consider a model where a finite number of producers compete to meet an infinitely divisible but inelastic demand for a product. Each firm is characterized by a production cost that is convex in the output produced, and firms act as profit maximizers. We consider a uniform price market design that uses supply function bidding: firms declare the amount they would supply at any positive price, and a single price is chosen to clear the market. We are interested in evaluating the impact of price-anticipating behavior both on the allocative efficiency of the market and on the prices seen at equilibrium. We show that by restricting the strategy space of the firms to parameterized supply functions, we can provide upper bounds on both the inflation of aggregate cost at the Nash equilibrium relative to the socially optimal level, as well as the markup of the Nash equilibrium price above the competitive level: as long as N > 2 firms are competing, these quantities are both upper bounded by 1 + 1/(N â 2). This result holds even in the presence of asymmetric cost structure across firms. We also discuss several extensions, generalizations, and related issues.National Science Foundation (U.S.) (Graduate Research Fellowship)National Science Foundation (U.S.) (grant ECS-0312921
Linearly Parameterized Bandits
We consider bandit problems involving a large (possibly infinite) collection
of arms, in which the expected reward of each arm is a linear function of an
-dimensional random vector , where .
The objective is to minimize the cumulative regret and Bayes risk. When the set
of arms corresponds to the unit sphere, we prove that the regret and Bayes risk
is of order , by establishing a lower bound for an
arbitrary policy, and showing that a matching upper bound is obtained through a
policy that alternates between exploration and exploitation phases. The
phase-based policy is also shown to be effective if the set of arms satisfies a
strong convexity condition. For the case of a general set of arms, we describe
a near-optimal policy whose regret and Bayes risk admit upper bounds of the
form .Comment: 40 pages; updated results and reference
Optimal scaling of average queue sizes in an input-queued switch: an open problem
We review some known results and state a few versions of an open problem related to the scaling of the total queue size (in steady state) in an nĂn input-queued switch, as a function of the port number n and the load factor Ï. Loosely speaking, the question is whether the total number of packets in queue, under either the maximum weight policy or under an optimal policy, scales (ignoring any logarithmic factors) as O(n/(1 â Ï)).National Science Foundation (U.S.) (Grant CCF-0728554
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