Auction algorithms for generalized nonlinear network flow problems

Thesis (Ph.D.)--Boston UniversityNetwork flow is an area of optimization theory concerned with optimization over networks with a range of applicability in fields such as computer networks, manufacturing, finance, scheduling and routing, telecommunications, and transportation. In both linear and nonlinear networks, a family of primal-dual algorithms based on "approximate" Complementary Slackness (ε-CS) is among the fastest in centralized and distributed environments. These include the auction algorithm for the linear assignment/transportation problems, ε-relaxation and Auction/Sequential Shortest Path (ASSP) for the min-cost flow and max-flow problems. Within this family, the auction algorithm is particularly fast, as it uses "second best" information, as compared to using the more generic ε-relaxation for linear assignment/transportation. Inspired by the success of auction algorithms, we extend them to two important classes of nonlinear network flow problems. We start with the nonlinear Resource Allocation Problem (RAP). This problem consists of optimally assigning N divisible resources to M competing missions/tasks each with its own utility function. This simple yet powerful framework has found applications in diverse fields such as finance, economics, logistics, sensor and wireless networks. RAP is an instance of generalized network (networks with arc gains) flow problem but it has significant special structure analogous to the assignment/transportation problem. We develop a class of auction algorithms for RAP: a finite-time auction algorithm for both synchronous and asynchronous environments followed by a combination of forward and reverse auction with ε-scaling to achieve pseudo polynomial complexity for any non-increasing generalized convex utilities including non-continuous and/ or non-differentiable functions. These techniques are then generalized to handle shipping costs on allocations. Lastly, we demonstrate how these techniques can be used for solving a dynamic RAP where nodes may appear or disappear over time. In later part of the thesis, we consider the convex nonlinear min-cost flow problem. Although E-relaxation and ASSP are among the fastest available techniques here, we illustrate how nonlinear costs, as opposed to linear, introduce a significant bottleneck on the progress that these algorithms make per iteration. We then extend the core idea of the auction algorithm, use of second best to make aggressive steps, to overcome this bottleneck and hence develop a faster version of ε-relaxation. This new algorithm shares the same theoretical complexity as the original but outperforms it in our numerical experiments based on random test problem suites

Multiple-Target Tracking in Complex Scenarios

In this dissertation, we develop computationally efficient algorithms for multiple-target tracking: MTT) in complex scenarios. For each of these scenarios, we develop measurement and state-space models, and then exploit the structure in these models to propose efficient tracking algorithms. In addition, we address design issues such as sensor selection and resource allocation. First, we consider MTT when the targets themselves are moving in a time-varying multipath environment. We develop a sparse-measurement model that allows us to exploit the inherent joint delay-Doppler diversity offered by the environment. We then reformulate the problem of MTT as a block-support recovery problem using the sparse measurement model. We exploit the structure of the dictionary matrix to develop a computationally efficient block support recovery algorithm: and thereby a multiple-target tracking algorithm) under the assumption that the channel state describing the time-varying multipath environment is known. Further, we also derive an upper bound on the overall error probability of wrongly identifying the support of the sparse signal. We then relax the assumption that the channel state is known. We develop a new particle filter called the Multiple Rao-Blackwellized Particle Filter: MRBPF) to jointly estimate both the target and the channel states. We also compute the posterior Cramér-Rao bound: PCRB) on the estimates of the target and the channel states and use the PCRB to find a suitable subset of antennas to be used for transmission in each tracking interval, as well as the power transmitted by these antennas. Second, we consider the problem of tracking an unknown number and types of targets using a multi-modal sensor network. In a multi-modal sensor network, different quantities associated with the same state are measured using sensors of different kinds. Hence, an efficient method that can suitably combine the diverse information measured by each sensor is required. We first develop a Hierarchical Particle Filter: HPF) to estimate the unknown state from the multi-modal measurements for a special class of problems which can be modeled hierarchically. We then model our problem of tracking using a hierarchical model and then use the proposed HPF for joint initiation, termination and tracking of multiple targets. The multi-modal data consists of the measurements collected from a radar, an infrared camera and a human scout. We also propose a unified framework for multi-modal sensor management that comprises sensor selection: SS), resource allocation: RA) and data fusion: DF). Our approach is inspired by the trading behavior of economic agents in commercial markets. We model the sensors and the sensor manager as economic agents, and the interaction among them as a double sided market with both consumers and producers. We propose an iterative double auction mechanism for computing the equilibrium of such a market. We relate the equilibrium point to the solutions of SS, RA and DF. Third, we address MTT problem in the presence of data association ambiguity that arises due to clutter. Data association corresponds to the problem of assigning a measurement to each target. We treat the data association and state estimation as separate subproblems. We develop a game-theoretic framework to solve the data association, in which we model each tracker as a player and the set of measurements as strategies. We develop utility functions for each player, and then use a regret-based learning algorithm to find the correlated equilibrium of this game. The game-theoretic approach allows us to associate measurements to all the targets simultaneously. We then use particle filtering on the reduced dimensional state of each target, independently

Gather-and-broadcast frequency control in power systems

We propose a novel frequency control approach in between centralized and distributed architectures, that is a continuous-time feedback control version of the dual decomposition optimization method. Specifically, a convex combination of the frequency measurements is centrally aggregated, followed by an integral control and a broadcast signal, which is then optimally allocated at local generation units. We show that our gather-and-broadcast control architecture comprises many previously proposed strategies as special cases. We prove local asymptotic stability of the closed-loop equilibria of the considered power system model, which is a nonlinear differential-algebraic system that includes traditional generators, frequency-responsive devices, as well as passive loads, where the sources are already equipped with primary droop control. Our feedback control is designed such that the closed-loop equilibria of the power system solve the optimal economic dispatch problem

Auctions, Bidding and Exchange Design

The different auction types are outlined using a classification framework along six dimensions. The economic properties that are desired in the design of auction mechanisms and the complexities that arise in their implementation are discussed. Some of the most interesting designs from the literature are analyzed in detail to establish known results and to identify the emerging research directions.Engineering and Applied Science