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14
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.
The Role of Linear SemiInfinite Programming in SignalAdapted QMF Bank Design
, 1995
"... The design of an orthogonal FIR quadraturemirror filter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a difficult nonlinear constrained optimization problem. In this paper, it is shown t ..."
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Cited by 19 (6 self)
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The design of an orthogonal FIR quadraturemirror filter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a difficult nonlinear constrained optimization problem. In this paper, it is shown that in fact the coding gain depends only upon the product filter P (z) = H(z)H(z \Gamma1 ), and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints, termed linear semi infinite programming (SIP) problems. The soughtfor, original filter, H(z), is obtained by deflation and spectral factorization of P (z). With the SIP formulation, every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms. The natural regularity properties inherent in the SIP formulation enhance the performance of these algorithms. We present a comprehensive theoretical analysis of ...
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...
Complexity Analysis Of A Logarithmic Barrier Decomposition Methods For SemiInfinite Linear Programming
, 1997
"... In this paper, we analyze a logarithmic barrier decomposition method for solving a semiinfinite linear programming problem. This method is in some respects similar to the column generation methods using analytic centers. Although the method was found to be very efficient in the recent computational ..."
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Cited by 10 (2 self)
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In this paper, we analyze a logarithmic barrier decomposition method for solving a semiinfinite linear programming problem. This method is in some respects similar to the column generation methods using analytic centers. Although the method was found to be very efficient in the recent computational studies, its theoretical convergence or complexity is still unknown except in the (finite) case of linear programming. In this paper we present a complexity analysis of this method in the general semiinfinite case. Our complexity estimate is given in terms of the problem dimension, the radius of the largest Euclidean ball contained in the feasible set, and the desired accuracy of the approximate solution. KEY WORDS. Semiinfinite linear programming, logarithmic barrier, decomposition, column generation. AMS subject classification: 90C25, 90C60. iii 1 Introduction Consider the following semiinfinite linear programming problem (SILP): maximize f 0 (y) := a T 0 y subject to f t (y) := ...
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...
Column Generation with a PrimalDual Method
, 1997
"... A simple column generation scheme that employs an interior point method to solve underlying restricted master problems is presented. In contrast with the classical column generation approach where restricted master problems are solved exactly, the method presented in this paper consists in solving i ..."
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Cited by 9 (2 self)
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A simple column generation scheme that employs an interior point method to solve underlying restricted master problems is presented. In contrast with the classical column generation approach where restricted master problems are solved exactly, the method presented in this paper consists in solving it to a predetermined optimality tolerance (loose at the beginning and appropriately tightened when the optimum is approached). An infeasible primaldual interior point method which employs the notion of ¯center to control the distance to optimality is used to solve the restricted master problem. Similarly to the analytic center cutting plane method, the present approach takes full advantage of the use of central prices. Furthermore, it offers more freedom in the choice of optimization strategy as it adaptively adjusts the required optimality tolerance in the master to the observed rate of convergence of the column generation process. The proposed method has been implemented and used to solv...
A long step cutting plane algorithm that uses the volumetric barrier
, 1995
"... A cutting plane method for linear/convex programming is described. It is based on the volumetric barrier, introduced by Vaidya. The algorithm is a long step one, and has a complexity of O(n1.5L) Newton steps. This is better than the O(n √ mL) complexity of noncutting plane long step methods based o ..."
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Cited by 8 (5 self)
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A cutting plane method for linear/convex programming is described. It is based on the volumetric barrier, introduced by Vaidya. The algorithm is a long step one, and has a complexity of O(n1.5L) Newton steps. This is better than the O(n √ mL) complexity of noncutting plane long step methods based on the volumetric barrier, but it is however worse than Vaidya’s original O(nL) result (which is not a long step algorithm). Major features of our algorithm are that when adding cuts we add them right through the current point, and when seeking progress in the objective, the duality gap is reduced by half (not provably true for Vaidya’s original algorithm). Further, we generate primal as well as dual iterates, making this applicable in the column generation context as well. Vaidya’s algorithm has been used as a subroutine to obtain the best complexity for several combinatorial optimization problems – e.g, the HeldKarp lower bound for the Traveling Salesperson Problem. While our complexity result is weaker, this long step cutting plane algorithm is likely to be computationally more promising on such combinatorial optimization problems with an exponential number of constraints. We also discuss a multiple cuts version — where upto p ≤ n ‘selectively orthonormalized ’ cuts are added through the current point. This has a complexity of O(n1.5Lp log p) quasi Newton steps.
A Proximal Cutting Plane Method Using Chebychev Center for Nonsmooth Convex Optimization
, 2006
"... An algorithm is developped for minimizing nonsmooth convex functions. This algortithm extends ElzingaMoore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to ElzingaMoore’s algorithm what ..."
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Cited by 5 (0 self)
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An algorithm is developped for minimizing nonsmooth convex functions. This algortithm extends ElzingaMoore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to ElzingaMoore’s algorithm what a proximal bundle method is to Kelley’s algorithm. As in proximal bundle methods, a quadratic problem is solved at each iteration, but the usual polyhedral approximation values are not used. We propose some variants and using some academic test problems, we conduct a numerical comparative study with three other nonsmooth methods.
Using the Primal Dual Infeasible Newton Method in the Analytic Center Method for Problems Defined by Deep Cutting Planes.
, 1998
"... The convergence and the complexity of a primaldual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a "deep cut" separation oracle is studied. The primaldual infeasible Newton method is used to generate a primaldual ..."
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Cited by 1 (0 self)
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The convergence and the complexity of a primaldual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a "deep cut" separation oracle is studied. The primaldual infeasible Newton method is used to generate a primaldual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is a cut that is tangent to the primaldual variant of Dikin's ellipsoid. Keywords: Convex feasibility problem, analytic center, column generation, cutting planes, deep cut. AMS subject classification: 90C25, 90C26, 90C60. 1 This research is supported by the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152, by the FCAR of Quebec and by an Obermann fellowship at the Center for Advanced Studies at the University of Iowa. 1 Introduction The convex feasibility problem defined by a separation oracle is: find an interior poin...