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29
Network Optimization: Continuous and Discrete Models, Athena Scientific
, 1998
"... All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. ..."
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Cited by 130 (9 self)
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All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.
Convex Nondifferentiable Optimization: A Survey Focussed On The Analytic Center Cutting Plane Method.
, 1999
"... We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct pr ..."
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Cited by 51 (2 self)
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We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
A Cutting Plane Method from Analytic Centers for Stochastic Programming
 Mathematical Programming
, 1994
"... The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition ..."
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Cited by 49 (18 self)
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The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization. 1 Introduction The study of optimization problems in the presence of uncertainty still taxes the limits of methodology and software. One of the most approachable settings is that of twostaged planning under uncertainty, in which a first stage decision has to be taken bef...
Solving Nonlinear Multicommodity Flow Problems By The Analytic Center Cutting Plane Method
, 1995
"... The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear prog ..."
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Cited by 29 (14 self)
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The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with the Dijkstra's dheap algorithm. An implementation is described that that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on wellknown nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities). This research has been supported by the Fonds National de la Recherche Scientifique Suisse, grant #12 \Gamma 34002:92, NSERCCanada and ...
Multiple Cuts in the Analytic Center Cutting Plane Method
, 1998
"... We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables wi ..."
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Cited by 26 (1 self)
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We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variancecovariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(p log(p + 1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix primal, dual or primaldual that is used in the computations. The computation of the optimal direction uses Newton's method applied to a selfconcordant function of p variab...
The Analytic Center Cutting Plane Method with Semidefinite Cuts
 SIAM JOURNAL ON OPTIMIZATION
, 2000
"... We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution s ..."
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Cited by 16 (1 self)
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We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution set, is a compact set consists of piecewise algebraic surfaces. We prove that the analytic center is recovered after adding a pdimensional cut in O(p log(p 1)) damped Newton's iteration. We also prove that the algorithm stops when the dimension of the accumulated block diagonal matrix cut reaches to the bound of O (p 2 m 3 =ffl 2 ), where p is the maximum dimension cut and ffl is radius of the largest ball contained in the solution set.
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.
A buildup variant of the pathfollowing method for LP
 OR Letters
, 1991
"... We propose a strategy for building up the linear program while using a logarithmic barrier method. The method starts with a (small) subset of the dual constraints, and follows the corresponding central path until the iterate is close to (or violates) one of the constraints, which is in turn added to ..."
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Cited by 12 (1 self)
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We propose a strategy for building up the linear program while using a logarithmic barrier method. The method starts with a (small) subset of the dual constraints, and follows the corresponding central path until the iterate is close to (or violates) one of the constraints, which is in turn added to the current system. This process is repeated until an optimal solution is reached. If a constraint is added to the current system, the central path will, of course, change. We analyze the effect on the barrier function value if a constraint is added. More importantly, we give an upper bound for the number of iterations needed to return to the new path. We prove that in the worst case the complexity is the same as that of the standard logarithmic barrier method. In practice this buildup scheme is likely to save a great deal of computation. Key Words: interior point method, linear programming, logarithmic barrier function, polynomial algorithm, buildup variant. 1 Introduction Karmarkar...
A long step cutting plane algorithm that uses the volumetric barrier
, 1995
"... A cutting plane method for linear/convex programming is described. It is based on the volumetric barrier, introduced by Vaidya. The algorithm is a long step one, and has a complexity of O(n1.5L) Newton steps. This is better than the O(n √ mL) complexity of noncutting plane long step methods based o ..."
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Cited by 8 (5 self)
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A cutting plane method for linear/convex programming is described. It is based on the volumetric barrier, introduced by Vaidya. The algorithm is a long step one, and has a complexity of O(n1.5L) Newton steps. This is better than the O(n √ mL) complexity of noncutting plane long step methods based on the volumetric barrier, but it is however worse than Vaidya’s original O(nL) result (which is not a long step algorithm). Major features of our algorithm are that when adding cuts we add them right through the current point, and when seeking progress in the objective, the duality gap is reduced by half (not provably true for Vaidya’s original algorithm). Further, we generate primal as well as dual iterates, making this applicable in the column generation context as well. Vaidya’s algorithm has been used as a subroutine to obtain the best complexity for several combinatorial optimization problems – e.g, the HeldKarp lower bound for the Traveling Salesperson Problem. While our complexity result is weaker, this long step cutting plane algorithm is likely to be computationally more promising on such combinatorial optimization problems with an exponential number of constraints. We also discuss a multiple cuts version — where upto p ≤ n ‘selectively orthonormalized ’ cuts are added through the current point. This has a complexity of O(n1.5Lp log p) quasi Newton steps.
Polynomial Cutting Plane Algorithms for TwoStage Stochastic Linear Programs Based on Ellipsoids, Volumetric Centers and Analytic Centers
 Washington State University
, 1995
"... Traditional simplexbased algorithms for twostage stochastic linear programscan be broadly divided into two classes: (a) those that explicitly exploit the structure of the equivalent largescale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly ..."
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Cited by 6 (3 self)
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Traditional simplexbased algorithms for twostage stochastic linear programscan be broadly divided into two classes: (a) those that explicitly exploit the structure of the equivalent largescale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly exploit that structure. Algorithms of class (b) are in general preferred. In 1988, following the work of Karmarkar for general linear programs, Birge and Qi [10] proposed a specialization of Karmarkar's algorithm for twostage stochastic linear programs. The algorithm of Birge and Qi [10] is the first interior point analog of class (a). Several other authors have studied related and different interior point analogs of class (a). Birge and Qi [10] also presented an analysis of the computational complexity of their algorithm. This analysis indicates that the computational complexity (in terms of total arithmetic operations) of their algorithm is in general smaller than that of the Karmarkar's ...