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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.
ACCPM  A Library for Convex Optimization Based on an Analytic Center Cutting Plane Method
 European Journal of Operational Research
, 1996
"... Introduction We are concerned in this note with the Goffin Haurie and Vial's [7] Analytic Center Cutting Plane Method (ACCPM for short) for largescale convex optimization. Its stateoftheart implementation [10] is now available upon request for academic research use. Cutting plane methods for co ..."
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Cited by 33 (17 self)
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Introduction We are concerned in this note with the Goffin Haurie and Vial's [7] Analytic Center Cutting Plane Method (ACCPM for short) for largescale convex optimization. Its stateoftheart implementation [10] is now available upon request for academic research use. Cutting plane methods for convex optimization have a long history that goes back at least to a fundamental paper of Kelley [14]. There exist numerous strategies that can be applied to "solve" subsequent relaxed master problems in the cutting planes optimization scheme. In the Analytic Center Cutting Plane Method, subsequent relaxed master problems are not solved to optimality. Instead of it, an approximate analytic center of the current localization set is looked for. The theoretical development of ACCPM started from Goffin and Vial [9]. It was later continued in [7, 8] and led to a development of the prototype implementation of the method due to du Merle [15] that was successfully applied to solve several nont
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...
Solving Linear Ordering Problems With A Combined Interior Point/simplex Cutting Plane Algorithm
, 1997
"... : We describe a cutting plane algorithm for solving linear ordering problems. The algorithm uses a primaldual interior point method to solve the first few relaxations and then switches to a simplex method to solve the last few relaxations. The simplex method uses CPLEX 4.0. We compare the algorithm ..."
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Cited by 26 (10 self)
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: We describe a cutting plane algorithm for solving linear ordering problems. The algorithm uses a primaldual interior point method to solve the first few relaxations and then switches to a simplex method to solve the last few relaxations. The simplex method uses CPLEX 4.0. We compare the algorithm with one that uses only an interior point method and with one that uses only a simplex method. We solve integer programming problems with as many as 31125 binary variables. Computational results show that the combined approach can dramatically outperform the other two methods. 1.1 INTRODUCTION The linear ordering problem has applications in economics, archaeology, scheduling, the social sciences, and aggregation of individual preferences. A cutting plane method provides a way to obtain a provably optimal solution to a linear ordering problem. Such a method requires the solution of a sequence of linear programming problems. It is now possible to solve linear ordering problems of a size wher...
Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities
, 1997
"... In this paper we consider a new analytic center cutting plane method in a projective space. We prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem, the variational inequality problem and the problem of constrained min ..."
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Cited by 20 (1 self)
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In this paper we consider a new analytic center cutting plane method in a projective space. We prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem, the variational inequality problem and the problem of constrained minimization. Our analysis is valid even for the problems whose solution belongs to the boundary of the domain. Keywords: Cutting plane, analytic centers. This research is partially supported by the Fonds National Suisse (grant # 1242503.94) 1 Introduction Cutting plane methods are designed to solve convex problems with the following property. A socalled oracle provides a first order information in the form of cutting planes that separate the query point from the set of solutions. Given a sequence of query points, the oracle answers a set of cutting planes that generates a polyhedral relaxation of the solution set. As the sequence of query points increases, the relaxation gets increasin...
A LogBarrier Method With Benders Decomposition For Solving TwoStage Stochastic Programs
 Mathematical Programming 90
, 1999
"... An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving twostage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite numb ..."
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Cited by 16 (6 self)
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An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving twostage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomialtime. Key Words: Stochastic programming, Largescale linear programming, Barrier function, Interior point methods, Benders decomposition, Complexity. Abbreviated Title: A logbarrier method with Benders decomposition AMS(MOS) subject classifications: 90C15, 90C05, 90C06, 90C60. 1 1. Introduction In this paper we propose an algorithm for solving twostage stochastic programs, establish fundamental properties of the algorithm, and analyze the convergence. An example of a twostage stochastic program is a production planning problem. The production and demand take place in the first and second periods, resp...
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 15 (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 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.
On the Comparative Behavior of Kelley's Cutting Plane Method and the Analytic Center Cutting plane Method
, 1996
"... In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method applied to convex optimization problems, which may lead to weaker results than Kelley's cutting plane method. Improvements to the analytic center cutting plane method are suggested. 1 Introd ..."
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Cited by 12 (8 self)
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In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method applied to convex optimization problems, which may lead to weaker results than Kelley's cutting plane method. Improvements to the analytic center cutting plane method are suggested. 1 Introduction In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method, and propose improvements. Cutting plane algorithms are designed to solve general convex optimization problems. They assume that the only information available around the current iterate takes the form of cutting planes, either supporting hyperplanes to the epigraph of the objective function, or separating hyperplanes from the feasible set. The two types of hyperplanes jointly define a linear programming, polyhedral, relaxation of the original convex optimization problem. The key issue in designing a specific cutting plane algorithm is the choice of a point in the current poly...