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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 16 (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
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.
Restoration of Services in Interdependent Infrastructure Systems: A Network Flows Approach
 Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute
, 2003
"... Abstract — Modern society depends on the operations of civil infrastructure systems, such as transportation, energy, telecommunications and water. Clearly, disruption of any of these systems would present a significant detriment to daily living. However, these systems have become so interconnected, ..."
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Cited by 7 (2 self)
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Abstract — Modern society depends on the operations of civil infrastructure systems, such as transportation, energy, telecommunications and water. Clearly, disruption of any of these systems would present a significant detriment to daily living. However, these systems have become so interconnected, one relying on another, that disruption of one may lead to disruptions in all. The focus of this research is on developing techniques which can be used to respond to events that have the capability to impact interdependent infrastructure systems. As discussed in the paper, infrastructure interdependencies occur when, due to either geographical proximity or shared operations, an impact on one infrastructure system affects one or more other infrastructure systems. The approach is to model the salient elements of these systems and provide decision makers with a means to manipulate the set of models, i.e. a decision support system. 1
Using selective orthonormalization to update the analytic center after the addition of multiple cuts
 Journal of Optimization Theory and Applications
"... We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every ‘stage ’ in a cutting plane algorithm. If q ≤ n cuts are to be added, we show that we can use a ‘Selec ..."
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Cited by 3 (2 self)
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We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every ‘stage ’ in a cutting plane algorithm. If q ≤ n cuts are to be added, we show that we can use a ‘Selective Orthonormalization ’ procedure to modify the cuts before adding them — it is then easy to identify a direction for an affine step into the interior of the new polytope, and the next analytic center is then found in O(q log q) Newton steps. Further, we show that multiple cut variants with selective orthonormalization of standard interior point cutting plane algorithms have the same complexity as the original algorithms.
Interior point methods for largescale linear programming
, 2004
"... We discuss interior point methods for largescale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primaldual interior point method, and consider the numerical issues involved in calculating the search direc ..."
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Cited by 1 (0 self)
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We discuss interior point methods for largescale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primaldual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned conjugate gradient methods. We also look at interior point column generation methods which can be used for very large scale linear programs or for problems where the data is generated only as needed.
A LongStep, Cutting Plane . . .
, 2000
"... A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya’s algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possib ..."
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A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya’s algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possible to maintain primal and dual iterates, thus allowing termination at will, instead of having to solve to completion. This algorithm has the same complexity (O(nL2) iterations) as Atkinson and Vaidya’s algorithm, but improves upon it in that it is a ‘longstep ’ version, while theirs is a ‘shortstep ’ one in some sense. For this reason, this algorithm is computationally much more promising as well. This algorithm can be of use in solving combinatorial optimization problems with large numbers of constraints, such as the Traveling Salesman Problem.
A LongStep, Cutting Plane Algorithm for Linear and Convex
, 1993
"... 1 Introduction The problem of interest is to solve min cT x s.t. Ax> = b (P) ..."
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1 Introduction The problem of interest is to solve min cT x s.t. Ax> = b (P)