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INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 15 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
A scaling algorithm for multicommodity flow problems
 Operations Research
, 1998
"... Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort to provide you with the best copy available. If you are dissatisfied with this product and find it unusable, please contact Document Services as soon as possible. ..."
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Cited by 10 (3 self)
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Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort to provide you with the best copy available. If you are dissatisfied with this product and find it unusable, please contact Document Services as soon as possible.
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.
Weighted Multidimensional Search and its Application to Convex Optimization
 SIAM J. COMPUT
, 1992
"... We present a weighted version of Megiddo's multidimensional search technique and use it to obtain faster algorithms for certain convex optimization problems in R d , for fixed d. This leads to speedups by a factor of log d n for applications such as solving the Lagrangian duals of matro ..."
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Cited by 8 (3 self)
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We present a weighted version of Megiddo's multidimensional search technique and use it to obtain faster algorithms for certain convex optimization problems in R d , for fixed d. This leads to speedups by a factor of log d n for applications such as solving the Lagrangian duals of matroidal knapsack problems and of constrained optimum subgraph problems on graphs of bounded treewidth.
Polynomial Cutting Plane Algorithms for TwoStage Stochastic Linear Programs Based on Ellipsoids, Volumetric Centers and Analytic Centers
 WASHINGTON STATE UNIVERSITY
, 1996
"... 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 ...
On the Stochastic Programming Discussed in D. Bertsimas and J.B. Orlin's Paper
"... Recently, Bertsimas and Orlin [1] have given polynomial complexity for several combinatorial optimization subproblems using Vaidya's volumetric center algorithm as a subroutine. These complexity results improve on the previously best known for several problems, including for example computing t ..."
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Recently, Bertsimas and Orlin [1] have given polynomial complexity for several combinatorial optimization subproblems using Vaidya's volumetric center algorithm as a subroutine. These complexity results improve on the previously best known for several problems, including for example computing the HeldKarp lower bound for Traveling Salesperson problem and Stochastic programming, and so on. In this paper, we point out some mistakes they made in their applications, especially in their applications on the Stochastic Programming. Key Words: Complexity, ellipsoid algorithm, volumetric center. 1 Introduction Bertsimas and Orlin [1] consider first the linear programs of special structure (P 1 ) z 1 = Min cx subject to Ax = b; (1) x 2 S; This research was supported in part by NSF Grant CCR9403892, and by funds provided by Washington State University in the form of a summer research assistantship to the second author. Results presented in this paper form part of the doctoral dissertation...
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(npmL) complexity of noncutting planelong step methods based on ..."
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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(npmL) complexity of noncutting planelong 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 whenadding 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 complexityfor 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 planealgorithm is likely to be computationally more promising on such combinatorial optimization problems with an exponential number of constraints. We also discuss a multiple cuts versionwhere upto p < = n `selectively orthonormalized ' cuts are added through the current point. Thishas a complexity of O(n1.5Lp log p) quasi Newton steps.
A note on a polynomial method for twostage stochastic linear programs proposed by Bertsimas and Orlin
"... We point out several oversights in statements concerning stochastic programming in a paper of Bertsimas and Orlin (this journal, 1994). Keywords: Stochastic programming, ellipsoid algorithm, Vaidya's algorithm, polynomial complexity. 1 Introduction In [2, x3.11], Bertsimas and Orlin state a me ..."
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We point out several oversights in statements concerning stochastic programming in a paper of Bertsimas and Orlin (this journal, 1994). Keywords: Stochastic programming, ellipsoid algorithm, Vaidya's algorithm, polynomial complexity. 1 Introduction In [2, x3.11], Bertsimas and Orlin state a method for a certain class of twostage stochastic linear programs and indicate a polynomial complexity result on the number of arithmetic operations required by their method to find an approximate solution. The purpose of this note is to point out several oversights in statements concerning stochastic programming contained in [2]. The rest of this note is structured as follows. The next section sketches the development of the method for stochastic programming given in [2]. It sets the stage for the concluding section in which we indicate several oversights in statements that Bertsimas and Orlin [2] make concerning stochastic programming. 1 This research was supported in part by NSF Grant CCR94038...