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Simultaneous Routing and Resource Allocation via Dual Decomposition
, 2004
"... In wireless data networks the optimal routing of data depends on the link capacities which, in turn, are determined by the allocation of communications resources (such as transmit powers and bandwidths) to the links. The optimal performance of the network can only be achieved by simultaneous optimi ..."
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Cited by 108 (4 self)
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In wireless data networks the optimal routing of data depends on the link capacities which, in turn, are determined by the allocation of communications resources (such as transmit powers and bandwidths) to the links. The optimal performance of the network can only be achieved by simultaneous optimization of routing and resource allocation. In this paper, we formulate the simultaneous routing and resource allocation problem and exploit problem structure to derive ef£cient solution methods. We use a capacitated multicommodity flow model to describe the data ¤ows in the network. We assume that the capacity of a wireless link is a concave and increasing function of the communications resources allocated to the link, and the communications resources for groups of links are limited. These assumptions allow us to formulate the simultaneous routing and resource allocation problem as a convex optimization problem over the network flow variables and the communications variables. These two sets of variables are coupled only through the link capacity constraints. We exploit this separable structure by dual decomposition. The resulting solution method attains the optimal coordination of data routing in the network layer and resource allocation in the radio control layer via pricing on the link capacities.
Selected topics in column generation
 Operations Research
, 2002
"... DantzigWolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual poin ..."
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Cited by 72 (5 self)
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DantzigWolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual point of view, which brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."
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 30 (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
Polynomial interior point cutting plane methods
 Optimization Methods and Software
, 2003
"... Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approxim ..."
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Cited by 15 (8 self)
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Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Longstep versions of the algorithms for solving convex optimization problems are presented. 1
A multiplecut analytic center cutting plane method for semidefinite feasibility problems
 SIAM Journal on Optimization
, 2002
"... form of these problems can be described as finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m × m symmetric positive semidefinite matrix ˆ Y either confirms that ˆ Y ∈ Γ or returns ..."
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Cited by 13 (3 self)
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form of these problems can be described as finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m × m symmetric positive semidefinite matrix ˆ Y either confirms that ˆ Y ∈ Γ or returns a cut, i.e., a symmetric matrix A such that Γ is in the halfspace {Y: A • Y ≤ A • ˆ Y}. We study an analytic center cutting plane algorithm for this problem. At each iteration the algorithm computes an approximate analytic center of a working set defined by the cuttingplane system generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise the new cutting plane returned by the oracle is added into the system. As the number of iterations increases, the working set shrinks and the algorithm eventually finds a solution of the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst case complexity of O ∗ (m 3 /ɛ 2) on the total number of cuts to be used, where ɛ is the maximum radius of a ball contained by Γ.
Building and Solving Largescale Stochastic Programs on an Affordable Distributed Computing System
, 1999
"... We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system the modeling language, the solver and the hardware are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial mod ..."
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Cited by 10 (1 self)
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We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system the modeling language, the solver and the hardware are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial model, which can be expanded to arbitrarily large sizes by enlarging the number of scenarios. We generated a model with one million scenarios, whose deterministic equivalent linear program has 1,111,112 constraints and 2,555,556 variables. We have been able to solve it on the cluster of ten PCs in less than 3 hours. Key words. Algebraic modeling language, decomposition methods, distributed systems, largescale optimization, stochastic programming. 1 Introduction Practical implementations of stochastic programming involve two big challenges. First, we have to build the model: its size is almost invariably large, if not huge, and this task is in itself a challenge for This research was suppor...
Higher order positive semidefinite diffusion tensor imaging
 SIAM J. Imaging Sci
"... Due to the wellknown limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) is used to characterize nonGaussian diffusion processes. One approach to analyze HARDI data is to model the apparent diffusion coefficient (ADC) with higher order diffusion tensors ..."
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Cited by 7 (1 self)
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Due to the wellknown limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) is used to characterize nonGaussian diffusion processes. One approach to analyze HARDI data is to model the apparent diffusion coefficient (ADC) with higher order diffusion tensors (HODT). The diffusivity function is positive semidefinite. In the literature, some methods have been proposed to preserve positive semidefiniteness of second order and fourth order diffusion tensors. None of them can work for arbitrary high order diffusion tensors. In this paper, we propose a comprehensive model to approximate the ADC profile by a positive semidefinite diffusion tensor of either second or higher order. We call this model PSDT (positive semidefinite diffusion tensor). PSDT is a convex optimization problem with a convex quadratic objective function constrained by the nonnegativity requirement on the smallest Zeigenvalue of the diffusivity function. The smallest Zeigenvalue is a computable measure of the extent of positive definiteness of the diffusivity function. We also propose some other invariants for the ADC profile analysis. Experiment results show that higher order tensors could improve the estimation of anisotropic diffusion and the PSDT model can
An interior point cutting plane method for the convex feasibility problem with secondorder cone inequalities
, 2004
"... ..."
A Proximal Cutting Plane Method Using Chebychev Center for Nonsmooth Convex Optimization
, 2006
"... An algorithm is developped for minimizing nonsmooth convex functions. This algortithm extends ElzingaMoore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to ElzingaMoore’s algorithm what ..."
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Cited by 5 (0 self)
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An algorithm is developped for minimizing nonsmooth convex functions. This algortithm extends ElzingaMoore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to ElzingaMoore’s algorithm what a proximal bundle method is to Kelley’s algorithm. As in proximal bundle methods, a quadratic problem is solved at each iteration, but the usual polyhedral approximation values are not used. We propose some variants and using some academic test problems, we conduct a numerical comparative study with three other nonsmooth methods.