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72
On the construction of some capacityapproaching coding schemes
, 2000
"... This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source ..."
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Cited by 82 (2 self)
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This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint sourcechannel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signaltonoise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on spacefilling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on spacefilling curves operates within 1.1 dB of Shannon’s ratedistortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the ratedistortion bound. The second scheme is based on lowdensity paritycheck (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution
Nonlinear optimal control via occupation measures and LMI relaxations
 SIAM Journal on Control and Optimization
, 2008
"... Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state an ..."
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Cited by 47 (24 self)
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Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (linear matrix inequality)relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments. 1.
Characterization and Computation of Optimal Distributions for Channel Coding
 IEEE Trans. Inform. Theory
, 2004
"... This paper concerns the structure of optimal codes for stochastic channel models. An investigation of an associated dual convex program reveals that the optimal distribution in channel coding is typically discrete. Based on this observation we obtain the following theoretical conclusions, as well as ..."
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Cited by 45 (3 self)
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This paper concerns the structure of optimal codes for stochastic channel models. An investigation of an associated dual convex program reveals that the optimal distribution in channel coding is typically discrete. Based on this observation we obtain the following theoretical conclusions, as well as new algorithms for constructing capacityachieving distributions: (i) Under general conditions, for low SNR the optimal random code is defined by a distribution whose magnitude is binary. (ii) Simple discrete approximations can nearly reach capacity even in cases where the optimal distribution is known to be absolutely continuous with respect to Lebesgue measure. (iii) A new class of algorithms is introduced, based on the cuttingplane method, to generate discrete distributions that are optimal within a prescribed class. Keywords: Information theory; channel coding; fading channels. # Department of Electrical and Computer Engineering, the Coordinated Science Laboratory, and the University of Illinois, 1308 W. Main Street, Urbana, IL 61801, URL http://black.csl.uiuc.edu:80/#meyn (smeyn@uiuc.edu). Work supported in part by the National Science Foundation through ITR 0085929 1
Optimal Auctions with Financially Constrained Bidders
, 2009
"... We consider an environment where potential exante symmetric buyers of an indivisible good have liquidity constraints, in that they cannot pay more than their ‘budget’ regardless of their valuation. A buyer’s valuation for the good as well as her budget are her private information. We derive the sym ..."
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Cited by 31 (4 self)
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We consider an environment where potential exante symmetric buyers of an indivisible good have liquidity constraints, in that they cannot pay more than their ‘budget’ regardless of their valuation. A buyer’s valuation for the good as well as her budget are her private information. We derive the symmetric constrainedefficient and revenue maximizing auctions for this setting. In general, the optimal auction requires ‘pooling’ both at the top and in the middle despite the maintained assumption of a monotone hazard rate. Further, the auctioneer will never find it desirable to subsidize bidders with low budgets.
Convex computation of the region of attraction of polynomial control systems
, 2012
"... We address the longstanding problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving ..."
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Cited by 24 (12 self)
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We address the longstanding problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finitedimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual LP on nonnegative continuous functions (approximated by polynomial sumofsquares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions.
Efficient ContinuousTime Dynamic Network Flow Algorithms
 OPERATIONS RESEARCH LETTERS
, 1998
"... We extend the discretetime dynamic flow algorithms presented in the literature to solve the analogous continuoustime dynamic flow problems. These problems include finding maximum dynamic flows, quickest flows, universally maximum dynamic flows, lexicographically maximum dynamic flows, dynamic tran ..."
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Cited by 24 (3 self)
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We extend the discretetime dynamic flow algorithms presented in the literature to solve the analogous continuoustime dynamic flow problems. These problems include finding maximum dynamic flows, quickest flows, universally maximum dynamic flows, lexicographically maximum dynamic flows, dynamic transshipments, and quickest transshipments in networks with capacities and transit times on the edges.
From fluid relaxations to practical algorithms for job shop scheduling: the makespan objective
 Mathematical Programming
, 2002
"... We design an algorithm for the highmultiplicity jobshop scheduling problem with the objective of minimizing the total holding cost by appropriately rounding an optimal solution to a fluid relaxation in which we replace discrete jobs with the flow of a continuous fluid. The algorithm solves the flu ..."
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Cited by 23 (4 self)
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We design an algorithm for the highmultiplicity jobshop scheduling problem with the objective of minimizing the total holding cost by appropriately rounding an optimal solution to a fluid relaxation in which we replace discrete jobs with the flow of a continuous fluid. The algorithm solves the fluid relaxation optimally and then aims to keep the schedule in the discrete network close to the schedule given by the fluid relaxation. If the number of jobs from each type grow linearly with N,then the algorithm is within an additive factor O�N � from the optimal (which scales as O�N 2�); thus,it is asymptotically optimal. We report computational results on benchmark instances chosen from the OR library comparing the performance of the proposed algorithm and several commonly used heuristic methods. These results suggest that for problems of moderate to high multiplicity,the proposed algorithm outperforms these methods,and for very high multiplicity the overperformance is dramatic. For problems of low to moderate multiplicity,however,the relative errors of the heuristic methods are comparable to those of the proposed algorithm,and the best of these methods performs better overall than the proposed method. Received December 1999; revisions received July 2000,September 2001; accepted September 2002. Subject classifications: Production/scheduling,deterministic: approximation algorithms for deterministic job shops. Queues,optimization: asymptotically optimal solutions to queueing networks. Area of review: Manufacturing,Service,and Supply Chain Operations. 1.
A Faster Algorithm for the Quickest Transshipment Problem
 In Proceedings of the Ninth Annual ACM/SIAM Symposium on Discrete Algorithms
, 1997
"... A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done using the minimum number of iterations? This is the question tackled in the quickest transshipment problem. Hoppe and Tardos [8] describe the only known polynomia ..."
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Cited by 19 (7 self)
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A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done using the minimum number of iterations? This is the question tackled in the quickest transshipment problem. Hoppe and Tardos [8] describe the only known polynomial time algorithm that finds an integral solution to this problem. Their algorithm repeatedly minimizes submodular functions using the ellipsoid method, and is therefore not at all practical. I present an algorithm that finds a fully integral quickest transshipment with a polynomial number of maximum flow computations. When there is only one sink, the quickest transshipment problem is significantly easier. For this case, I show how the algorithm can be sped up to return an integral solution using O(k) maximum flow computations, where k is the number of sources. Hajek and Ogier [7] describe an algorithm that finds a fractional solution to the singlesink quickest transshipment problem o...
The achievable region method in the optimal control of queueing systems; formulations, bounds and policies
 QUEUEING SYST
, 1995
"... We survey a new approach that the author and his coworkers have developed to formulate stochastic control problems (predominantly queueing systems) as mathematicalprogramming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., fi ..."
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Cited by 15 (4 self)
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We survey a new approach that the author and his coworkers have developed to formulate stochastic control problems (predominantly queueing systems) as mathematicalprogramming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable systems (multiclass single station queues and multiarmed bandit problems), restless bandit problems, polling systems, multiclass queueing and loss networks. These relaxations lead to bounds on the performance of an optimal policy. Using information from the relaxations we construct heuristic nearly optimal policies, The theme in the paper is the thesis that better formulations lead to deeper understanding and better solution methods. Overall the proposed approach for stochastic control problems parallels efforts of the mathematical programming community in the last twenty years to develop sharper formulations (polyhedral combinatorics and more recently nonlinear relaxations) and leads to new insights ranging from a complete characterization and new algorithms for indexable systems to tight lower bounds and nearly optimal algorithms for restless bandit problems, polling systems, multiclass queueing and loss networks.