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42
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...
Reoptimization with the PrimalDual Interior Point Method
, 2001
"... Reoptimization techniques for an interior point method applied to solve a sequence of linear programming problems are discussed. Conditions are given for problem perturbations that can be absorbed in merely one Newton step. The analysis is performed for both shortstep and longstep feasible pathf ..."
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Cited by 25 (10 self)
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Reoptimization techniques for an interior point method applied to solve a sequence of linear programming problems are discussed. Conditions are given for problem perturbations that can be absorbed in merely one Newton step. The analysis is performed for both shortstep and longstep feasible pathfollowing method. A practical procedure is then derived for an infeasible pathfollowing method. It is applied in the context of crash start for several largescale structured linear programs. Numerical results with OOPS, the new objectoriented parallel solver demonstrate the efficiency of the approach. For large structured linear programs crash start leads to about 40% reduction of the iterations number and translates into 25% reduction of the solution time. The crash procedure parallelizes well and speedups between 3.13.8 on 4 processors are achieved.
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.
Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely comb ..."
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Cited by 20 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
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.