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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...
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...
The Analytic Center Cutting Plane Method with Semidefinite Cuts
 SIAM JOURNAL ON OPTIMIZATION
, 2000
"... We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution s ..."
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Cited by 16 (1 self)
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We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution set, is a compact set consists of piecewise algebraic surfaces. We prove that the analytic center is recovered after adding a pdimensional cut in O(p log(p 1)) damped Newton's iteration. We also prove that the algorithm stops when the dimension of the accumulated block diagonal matrix cut reaches to the bound of O (p 2 m 3 =ffl 2 ), where p is the maximum dimension cut and ffl is radius of the largest ball contained in the solution set.
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...
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...
Survivability In Telecommunication Networks
, 1995
"... It is proposed that traffic in a telecommunications network be secured in the event of a node or link failure by the rerouting of traffic over a reserve network. The problem consists of two related parts: the dimensioning of a reserve network, and the reallocation, or rerouting of traffic. We formu ..."
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Cited by 8 (3 self)
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It is proposed that traffic in a telecommunications network be secured in the event of a node or link failure by the rerouting of traffic over a reserve network. The problem consists of two related parts: the dimensioning of a reserve network, and the reallocation, or rerouting of traffic. We formulate the problem as a linear programming problem of huge size which we solve using a cutting plane algorithm based on the concept of an analytic center. The method enables the solution of the survivability problem for networks with up to 60 nodes and 120 links, which allows a realistic modelling of France Telecom's Main Interconnection Network. Key words: Survivability in telecommunication networks, cutting plane methods, interior point methods, decomposition. FRANCE TELECOM, CNET, 3840, Rue du General Leclerc, 92131, Issy les Moulineaux Cedex, France. y LOGILAB, HECGen`eve, Universit'e de Gen`eve, 102 Bd Carl Vogt, CH1211 Gen`eve 4, Suisse. z This work was financed by contract No 9...
Interior Point Methods for Nondifferentiable Optimization
, 1998
"... We describe the analytic center cutting plane method and its relationship to classical methods of nondifferentiable optimization and column generation. Implementation issues are also discussed, and current applications listed. Keywords Projective Algorithm, Analytic Center, Cutting Plane Method. T ..."
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Cited by 6 (2 self)
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We describe the analytic center cutting plane method and its relationship to classical methods of nondifferentiable optimization and column generation. Implementation issues are also discussed, and current applications listed. Keywords Projective Algorithm, Analytic Center, Cutting Plane Method. This work has been completed with support from the Fonds National Suisse de la Recherche Scientifique, grant 1242503.94, from the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152 and from the FCAR of Quebec. GERAD/Faculty of Management, McGill University, 1001, Sherbrooke West, Montreal, Que., H3A 1G5, Canada. Email: ma56@musica.mcgill.ca. LOGILAB/Management Studies, University of Geneva, 102, Bd CarlVogt, CH1211 Gen`eve 4, Switzerland. Email: jpvial@hec.unige.ch. 1 Introduction Nondifferentiable convex optimization may be deemed an arcane topic in the field of optimization. Truly enough, many a times problems that are formulated as nondiffere...
A secondorder cone cutting surface method: complexity and application
, 2005
"... We present an analytic center cutting surface algorithm that uses mixed linear and multiple secondorder cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can ..."
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Cited by 6 (5 self)
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We present an analytic center cutting surface algorithm that uses mixed linear and multiple secondorder cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding p secondorder cone