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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 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...
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
InfeasibleStart PrimalDual Methods And Infeasibility Detectors For Nonlinear Programming Problems
 Mathematical Programming
, 1996
"... In this paper we present several "infeasiblestart" pathfollowing and potentialreduction primaldual interiorpoint methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a selfdual homogeneous primaldual problem. The methods under considerat ..."
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Cited by 31 (5 self)
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In this paper we present several "infeasiblestart" pathfollowing and potentialreduction primaldual interiorpoint methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a selfdual homogeneous primaldual problem. The methods under consideration generate an fflsolution for an fflperturbation of an initial strictly (primal and dual) feasible problem in O( p ln fflae f ) iterations, where is the parameter of a selfconcordant barrier for the cone, ffl is a relative accuracy and ae f is a feasibility measure. We also discuss the behavior of pathfollowing methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O( p ln ae \Delta ) iterations, where ae \Delta is a primal or dual infeasibility measure. 1 Introduction Nesterov and Nemirovskii [9] first developed and investigated extensions of several classes of interiorpoint algorithms for linear programming t...
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 ...
A Survey of Algorithms for Convex Multicommodity Flow Problems
, 1997
"... There are many problems related to the design of networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodit ..."
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Cited by 16 (2 self)
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There are many problems related to the design of networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of largescale convex optimization problems. We conduct some numerical experiments on the message routing problem with some different techniques. 1 Introduction The literature dealing with multicommodity flow problems is rich since the publication of the works of Ford and Fulkerson's [19] and T.C. Hu [30] in the beginning of the 1960s. These problems usually have a very large number of variables and constraints and arise in a great variety o...
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...
Using an Interior Point Method for the Master Problem in a Decomposition Approach
 European Journal of Operational Research
, 1997
"... We addres some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concern the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The parti ..."
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Cited by 11 (7 self)
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We addres some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concern the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The particular structure is the presence of GUB constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems. The method can be used in a fairly general case, i.e., in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections. Computational results demonstrate its advantages for one particular decomposition approach: Analytic Center Cutting Plane Method (ACCPM) is applied to solve large scale nonlinear multicommodity network flow problems (up to 5000 arcs and 10000 commodities). Key words. Convex programming, interior point methods, cutt...
Optimal Joint Synthesis of Base and Reserve Telecommunication Networks
, 1995
"... A telecommunication network is survivable if, following an arc failure, the interrupted traffic can be redirected through the network via existing excess capacity. The standard survivability problem consists in finding the least cost investment in spare capacity to allow rerouting of a given base tr ..."
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Cited by 9 (2 self)
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A telecommunication network is survivable if, following an arc failure, the interrupted traffic can be redirected through the network via existing excess capacity. The standard survivability problem consists in finding the least cost investment in spare capacity to allow rerouting of a given base traffic. In this paper we consider the more involved problem of simultaneously designing the base traffic and the spare capacity investment. If the investment costs are linear, the problem can be formulated as a large scale structured linear program that we solve using different decomposition techniques, including the analytic center cutting plane method. The global analysis is performed under the assumption of local rerouting of the traffic, i.e., the interrupted traffic creates a local demand between the end points of the broken edge. More sophisticated telecommunication network management allows to break down the interrupted traffic into its individual demand components. We do not treat the...