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 α\alpha-fair policies in bandwidth-sharing networks

We consider a flow-level model of a network operating under an $\alpha$-fair bandwidth sharing policy (with $\alpha>0$) 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 $\alpha\geq1$, 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 $\alpha\geq1$. For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any $\alpha>0$, 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 $\alpha=1$ 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 $N$ 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 $N$ 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 − 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 $r$-dimensional random vector $\mathbf{Z} \in \mathbb{R}^r$, where $r \geq 2$. 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 $\Theta(r \sqrt{T})$, 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 $O(r \sqrt{T} \log^{3/2} T)$.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 n×n input-queued switch, as a function of the port number n and the load factor ρ. 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 − ρ)).National Science Foundation (U.S.) (Grant CCF-0728554