Results 1  10
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16
Lectures on modern convex optimization
"... Mathematical Programming deals with optimization programs of the form and includes the following general areas: minimize f(x) subject to gi(x) ≤ 0, i = 1,..., m, [x ⊂ R n] 1. Modelling: methodologies for posing various applied problems as optimization programs; 2. Optimization Theory, focusing on e ..."
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Cited by 145 (6 self)
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Mathematical Programming deals with optimization programs of the form and includes the following general areas: minimize f(x) subject to gi(x) ≤ 0, i = 1,..., m, [x ⊂ R n] 1. Modelling: methodologies for posing various applied problems as optimization programs; 2. Optimization Theory, focusing on existence, uniqueness and on characterization of optimal solutions to optimization programs; 3. Optimization Methods: development and analysis of computational algorithms for various classes of optimization programs; 4. Implementation, testing and application of modelling methodologies and computational algorithms. Essentially, Mathematical Programming was born in 1948, when George Dantzig has invented Linear Programming – the class of optimization programs (P) with linear objective f(·) and
Interiorpoint methods for optimization
, 2008
"... This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 18 (0 self)
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This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
Joint Resource Allocation and User Association for Heterogeneous Wireless Cellular Networks
, 2012
"... We propose a unified static framework to study the interplay of user association and resource allocation in heterogeneous cellular networks. This framework allows us to compare the performance of three channel allocation strategies: Orthogonal deployment, Cochannel deployment, and Partially Shared ..."
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Cited by 16 (5 self)
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We propose a unified static framework to study the interplay of user association and resource allocation in heterogeneous cellular networks. This framework allows us to compare the performance of three channel allocation strategies: Orthogonal deployment, Cochannel deployment, and Partially Shared deployment. We have formulated joint optimization problems that are nonconvex integer programs, are NPhard, and hence it is difficult to efficiently obtain exact solutions. We have, therefore, developed techniques to obtain upper bounds on the system’s performance. We show that these upper bounds are tight by comparing them to feasible solutions. We have used these upper bounds as benchmarks to quantify how well different user association rules and resource allocation schemes perform. Our numerical results indicate that significant gains in throughput are achievable for heterogeneous networks if the right combination of user association and resource allocation is used. Noting the significant impact of the association rule on the performance, we propose a simple association rule that performs much better than all existing user association rules.
Cooperative profit sharing in coalition based resource allocation in wireless networks
 in Proc. of IEEE INFOCOM, (Rio de Janeiro
, 2009
"... Abstract—We consider a network in which several service providers offer wireless access service to their respective subscribed customers through potentially multihop routes. If providers cooperate, i.e., pool their resources, such as spectrum and base stations, and agree to serve each others ’ cust ..."
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Cited by 9 (4 self)
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Abstract—We consider a network in which several service providers offer wireless access service to their respective subscribed customers through potentially multihop routes. If providers cooperate, i.e., pool their resources, such as spectrum and base stations, and agree to serve each others ’ customers, their aggregate payoffs, and individual shares, can potentially substantially increase through efficient utilization of resources and statistical multiplexing. The potential of such cooperation can however be realized only if each provider intelligently determines who it would cooperate with, when it would cooperate, and how it would share its resources during such cooperation. Also, when the providers share their aggregate revenues, developing a rational basis for such sharing is imperative for the stability of the coalitions. We model such cooperation using transferable payoff coalitional game theory. We first consider the scenario that locations of the base stations and the channels that each provider can use have already been decided apriori (spectrum pooling game). We show that the optimum cooperation strategy, which involves the allocations of the channels and the base stations to mobile customers, can be obtained as solutions of convex optimizations. We next show that if all providers cooperate, there is always an operating point that maximizes the providers’ aggregate payoff, while offering each such a share that removes any incentive to split from the coalition. Next, we show that when the providers can choose the locations of their base stations and decide which channels to acquire, the above results hold in important special cases. Finally, we examine cooperation when providers do not share their payoffs, but still share their resources so as to enhance individual payoffs. We show that, in the spectrum pooling game, if all providers cooperate, there is always a joint action that fetches payoffs such that no subset of providers would break away from the coalition. I.
SemiDefinite Problems in Truss Topology Optimization
, 1995
"... In this report we review optimization problems arising from truss topology design, which can be formulated as positive semidefinite problems (PSP's). This is done with a view towards applying primaldual interior point methods for PSP's to obtain efficient nonlinear solvers for this class ..."
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Cited by 7 (2 self)
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In this report we review optimization problems arising from truss topology design, which can be formulated as positive semidefinite problems (PSP's). This is done with a view towards applying primaldual interior point methods for PSP's to obtain efficient nonlinear solvers for this class of problems.
"Conefree" primaldual pathfollowing and potential reduction polynomial time interiorpoint methods
 MATH. PROG
, 2005
"... We present a framework for designing and analyzing primaldual interiorpoint methods for convex optimization. We assume that a selfconcordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techni ..."
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Cited by 2 (2 self)
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We present a framework for designing and analyzing primaldual interiorpoint methods for convex optimization. We assume that a selfconcordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interiorpoint methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primaldual interiorpoint techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.
Randomized interior point methods for sampling and optimization
, 2009
"... We present a Markov chain (Dikin walk) for sampling from a convex body equipped with a selfconcordant barrier, whose mixing time from a “central point ” is strongly polynomial in the description of the convex set. The mixing time of this chain is invariant under affine transformations of the convex ..."
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Cited by 1 (1 self)
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We present a Markov chain (Dikin walk) for sampling from a convex body equipped with a selfconcordant barrier, whose mixing time from a “central point ” is strongly polynomial in the description of the convex set. The mixing time of this chain is invariant under affine transformations of the convex set, thus eliminating the need for first placing the body in an isotropic position. This strengthens and previous results of [11] for polytopes and generalizes these results to arbitrary convex sets. In the case of a convex set K defined by a semidefinite constraint of rank at most α and at most m additional linear constraints, our results specialize to the following statement. Let s ≥ pq  for any chord pq of K passing through a point x ∈ K. Then, after t = O n(m+ nα) n ln((m+ nα)s) + ln
DiSCO: Distributed Optimization for SelfConcordant Empirical Loss
"... We propose a new distributed algorithm for empirical risk minimization in machine learning. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and comm ..."
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We propose a new distributed algorithm for empirical risk minimization in machine learning. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing selfconcordant empirical loss functions, and discuss the results for distributed ridge regression, logistic regression and binary classification with a smoothed hinge loss. In a standard setting for supervised learning, where the n data points are i.i.d. sampled and when the regularization parameter scales as 1/ n, we show that the proposed algorithm is communication efficient: the required round of communication does not increase with the sample size n, and only grows slowly with the number of machines. 1.
Local Minimax Learning of Approximately Polynomial Functions
"... Suppose we have a number of noisy measurements of an unknown realvalued function f near a point of interest x0 ∈ Rd. Suppose also that nothing can be assumed about the noise distribution, except for zero mean and bounded covariance matrix. We want to estimate f at x0 using a general linear parametr ..."
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Suppose we have a number of noisy measurements of an unknown realvalued function f near a point of interest x0 ∈ Rd. Suppose also that nothing can be assumed about the noise distribution, except for zero mean and bounded covariance matrix. We want to estimate f at x0 using a general linear parametric family f(x; a) = a0h0(x) +... + aqhq(x), where a ∈ Rq and hi’s are bounded
SelfConcordant Barriers for Convex Approximations of Structured Convex Sets
, 2007
"... We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) selfconcordant barriers. This approach extends the reach of the modern theory of interiorpoint ..."
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We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) selfconcordant barriers. This approach extends the reach of the modern theory of interiorpoint methods, and lays the foundation for new ways to treat structured convex optimization problems with a very large number of constraints. Moreover, our approach provides a strong connection from the theory of selfconcordant barriers to the combinatorial optimization literature on solving packing and covering problems.