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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 Truncated PrimalInfeasible DualFeasible Network Interior Point Method
, 1994
"... . In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is ..."
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Cited by 29 (3 self)
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. In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code, pdnet, is tested on a large set of standard minimum cost network flow test problems. Computational results indicate that the implementation is competitive with stateoftheart network flow codes. Key Words. Interior point method, linear programming, network flows, primalinfeasible dualfeasible, truncated Newton method, conjugate gradient, maximum flow, experimental test...
Solving RealWorld Linear Ordering Problems . . .
, 1995
"... Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear prog ..."
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Cited by 21 (8 self)
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Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear programming relaxations. A point which is a good warm start for a simplexbased cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some realworld problems; the algorithm appears to be competitive with a simplexbased cutting plane algorithm.
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 14 (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.
Logarithmic Barrier Decomposition Methods for SemiInfinite Programming
, 1996
"... A computational study of some logarithmic barrier decomposition algorithms for semiinfinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hertog et al., to solv ..."
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Cited by 9 (1 self)
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A computational study of some logarithmic barrier decomposition algorithms for semiinfinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hertog et al., to solve semiinfinite programming problems. Usually decomposition (cutting plane methods) use cutting planes to improve the localization of the given problem. In this paper we propose an extension which uses linear cuts to solve large scale, difficult real world problems. This algorithm uses both static and (doubly) dynamic enumeration of the parameter space and allows for multiple cuts to be simultaneously added for larger/difficult problems. The algorithm is implemented both on sequential and parallel computers. Implementation issues and parallelization strategies are discussed and encouraging computational results are presented. Keywords: column generation, convex programming, cutting plane met...
Interior Point Algorithms For Network Flow Problems
 in Advances in linear and integer programming
, 1996
"... . Computational algorithms for the solution of network flow problems are of great practical significance. In the last decade, a new class of computationally efficient algorithms, based on the interior point method, has been proposed and applied to solve large scale network flow problems. In this cha ..."
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Cited by 8 (2 self)
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. Computational algorithms for the solution of network flow problems are of great practical significance. In the last decade, a new class of computationally efficient algorithms, based on the interior point method, has been proposed and applied to solve large scale network flow problems. In this chapter, we review interior point approaches for network flows, with emphasis on computational issues. Key words. Network flow problems, interior point methods, computational testing, computer implementation. AMS(MOS) subject classifications. 90B10, 90C05, 90C06, 90C35, 6505, 65F10, 65F50 1. Introduction. A large number of problems in transportation, communications, and manufacturing can be modeled as network flow problems. In these problems one seeks to find the most efficient, or optimal, way to move flow (e.g. materials, information, buses, electrical currents) on a network (e.g. postal network, computer network, transportation grid, power grid). Among these optimization problems, many a...
An Efficient Implementation of a Network Interior Point Method
, 1992
"... . We describe dlnet, an implementation of the dual affine scaling algorithm for minimum cost capacitated network flow problems. The efficiency of this implementation is the result of three factors: the small number of iterations taken by interior point methods, efficient solution of the linear syste ..."
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Cited by 8 (2 self)
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. We describe dlnet, an implementation of the dual affine scaling algorithm for minimum cost capacitated network flow problems. The efficiency of this implementation is the result of three factors: the small number of iterations taken by interior point methods, efficient solution of the linear system that determines the ascent direction using a preconditioned conjugate gradient algorithm and strategies to produce an optimal primal integer solution. The combination of these ingredients results in a code that can solve minimum cost network flow problems having over 250,000 vertices in a few hours of running time on a workstationclass computer. We compare dlnet with network simplex code netflo and relaxation code relaxt3 on an extensive range of minimum cost network flow problems, including minimum cost circulation, maximum flow and transshipment problems. The computational results show that dlnet offers more predictable running times than those of netflo and relaxt3. Its performance,...
Interior point algorithms for integer programming
 Advances in Linear and Integer Programming, chapter 6
, 1996
"... Research on using interior point algorithms to solve integer programming problems is surveyed. This paper concentrates on branch and bound and cutting plane methods; a potential function method is also briefly mentioned. The principal difficulty with using an interior point algorithm in a branch a ..."
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Cited by 7 (4 self)
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Research on using interior point algorithms to solve integer programming problems is surveyed. This paper concentrates on branch and bound and cutting plane methods; a potential function method is also briefly mentioned. The principal difficulty with using an interior point algorithm in a branch and cut method to solve integer programming problems is in warm starting the algorithm efficiently. Methods for overcoming this difficulty are described and other features of the algorithms are given. This paper focuses on the techniques necessary to obtain an efficient computational implementation; there is a short discussion of theoretical issues.
A constraintreduced variant of Mehrotra’s predictorcorrector algorithm. submitted for publication
 In Preparation
, 2007
"... Consider linear programs in dual standard form with n constraints and m variables. When typical interiorpoint algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires O(nm 2) ..."
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Cited by 6 (5 self)
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Consider linear programs in dual standard form with n constraints and m variables. When typical interiorpoint algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires O(nm 2) operations. When n ≫ m it is often the case that at each iteration most of the constraints are not very relevant for the construction of a good update and could be ignored to achieve computational savings. This idea has been considered in the 1990s by Dantzig and Ye, Tone, Kaliski and Ye, den Hertog et al. and others. More recently, Tits et al. proposed a simple “constraintreduction ” scheme and proved global and local quadratic convergence for a dualfeasible primaldual affinescaling method modified according to that scheme. In the present work, similar convergence results are proved for a dualfeasible constraintreduced variant of Mehrotra’s predictorcorrector algorithm. Some promising numerical results are reported. 1
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, 2010
"... 45 46 47 A constraintreduced variant of Mehrotra’s predictorcorrector algorithm ..."
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45 46 47 A constraintreduced variant of Mehrotra’s predictorcorrector algorithm