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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 157 (6 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.
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 42 (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
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 28 (17 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
Support Vector Machine Classification with Indefinite Kernels
, 2009
"... We propose a method for support vector machine classification using indefinite kernels. Instead of directly minimizing or stabilizing a nonconvex loss function, our algorithm simultaneously computes support vectors and a proxy kernel matrix used in forming the loss. This can be interpreted as a pena ..."
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Cited by 24 (1 self)
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We propose a method for support vector machine classification using indefinite kernels. Instead of directly minimizing or stabilizing a nonconvex loss function, our algorithm simultaneously computes support vectors and a proxy kernel matrix used in forming the loss. This can be interpreted as a penalized kernel learning problem where indefinite kernel matrices are treated as noisy observations of a true Mercer kernel. Our formulation keeps the problem convex and relatively large problems can be solved efficiently using the projected gradient or analytic center cutting plane methods. We compare the performance of our technique with other methods on several standard data sets.
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 22 (9 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 quasiconvex optimization approach to parameterized model order reduction
 In IEEE Proc. on Design Automation Conference
, 2005
"... Abstract—In this paper, an optimizationbased model order reduction (MOR) framework is proposed. The method involves setting up a quasiconvex program that solves a relaxation of the optimal H ∞ norm MOR problem. The method can generate guaranteed stable and passive reduced models and is very flexi ..."
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Cited by 21 (4 self)
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Abstract—In this paper, an optimizationbased model order reduction (MOR) framework is proposed. The method involves setting up a quasiconvex program that solves a relaxation of the optimal H ∞ norm MOR problem. The method can generate guaranteed stable and passive reduced models and is very flexible in imposing additional constraints such as exact matching of specific frequency response samples. The proposed optimizationbased approach is also extended to solve the parameterized modelreduction problem (PMOR). The proposed method is compared to existing moment matching and optimizationbased MOR methods in several examples. PMOR models for large RF inductors over substrate and powerdistribution grid are also constructed. Index Terms—Parameterized model order reduction (PMOR), quasiconvex optimization, RF inductor. I.
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 14 (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.
Comparison of Bundle and Classical Column Generation
"... When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cuttingplane algor ..."
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Cited by 13 (0 self)
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When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cuttingplane algorithm of Kelley and CheneyGoldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multiitem lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.