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Asymptotic Analysis of Congested Communication Networks
, 1997
"... : This paper is devoted to the study of a routing problem in telecommunication networks, when the cost function is the average delay. We establish asymptotic expansions for the value function and solutions in the vicinity of a congested nominal problem. The study is strongly related to the one of a ..."
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: This paper is devoted to the study of a routing problem in telecommunication networks, when the cost function is the average delay. We establish asymptotic expansions for the value function and solutions in the vicinity of a congested nominal problem. The study is strongly related to the one of a partial inverse barrier method for linear programming. Keywords: Telecommunication networks, multicommodity flows, asymptotic expansions, linear programming, perturbation analysis, barrier functions, penalty methods. (R'esum'e : tsvp) INRIA, B.P. 105, 78153 Rocquencourt, France. Email: Frederic.Bonnans@inria.fr. y INRIA, B.P. 105, 78153 Rocquencourt, France. Email: Mounir.Haddou@inria.fr. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Téléphone : (33 1) 39 63 55 11  Télécopie : (33 1) 39 63 53 Analyse asymptotique des r'eseaux de communications congestionn'es R'esum'e : Nous 'etudions le probl`eme de minimisation d...
Inverse barriers and CESfunctions in linear programming
, 1995
"... Recently much attention was paid to polynomial interior point methods, almost exclusively based on the logarithmic barrier function. Some attempts were made to prove polynomiality of other barrier methods (e.g. the inverse barrier method) but without success. Other interior point methods could be de ..."
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Recently much attention was paid to polynomial interior point methods, almost exclusively based on the logarithmic barrier function. Some attempts were made to prove polynomiality of other barrier methods (e.g. the inverse barrier method) but without success. Other interior point methods could be defined based on CESfunctions (CES is the abbreviation of Constant Elasticity of Substitution). The classical inverse barrier function and the CESfunctions have a similar structure. In this paper we compare the path defined by the inverse barrier function and the path defined by CESfunctions in the case of linear programming. It will be shown that the two paths are equivalent, although parameterized differently. We also construct a dual of the CESfunction problem which is based on the dual CESfunction. This result also completes the duality results for linear programs with one CEStype (pnorm) type constraint. Key words: linear programming, interiorpoint methods, inverse barrier, CESfunc...
The primal power affine scaling method
"... In this paper, we present a variant of the primal affine scaling method, which we call the primal power affine scaling method. This method is defined by choosing a real r> 0.5, and is similar to the power barrier variant of the primaldual homotopy methods considered by den Hertog, Roos and Terlaky ..."
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In this paper, we present a variant of the primal affine scaling method, which we call the primal power affine scaling method. This method is defined by choosing a real r> 0.5, and is similar to the power barrier variant of the primaldual homotopy methods considered by den Hertog, Roos and Terlaky and Sheu and Fang. Here, we analyze the methods for r> 1. The analysis for 0.50 < r < 1 is similar, and can be readily carried out with minor modifications. Under the nondegeneracy assumption, we show that the method converges for any choice of the step size a. To analyze the convergence without the nondegeneracy assumption, we define a power center of a polytope. We use the connection of the computation of the power center by Newton's method and the steps of the method to generalize the 2/3rd result of Tsuchiya and Muramatsu. We show that with a constant step size a such that c~/(1 tx) 2r < 2/(2r 1) and with a variable asymptotic step size ot k uniformly bounded away from 2/(2r + 1), the primal sequence converges to the relative interior of the optimal primal face, and the dual sequence converges to the power center of the optimal dual face. We also present an accelerated version of the method. We show that the twostep superlinear convergence rate of the method is 1 +r/(r+ 1), while the threestep convergence rate is 1 + 3r/(r + 2). Using the measure of Ostrowski, we note that the threestep method for r = 4 is more efficient than the twostep quadratically convergent method, which is the limit of the twostep method as r approaches infinity.
A MAJORIZEMINIMIZE LINE SEARCH ALGORITHM FOR BARRIER FUNCTION OPTIMIZATION
"... Many signal and image estimation problems such as maximum entropy reconstruction and positron emission tomography, require the minimization of a criterion containing a barrier function i.e., an unbounded function at the boundary of the feasible solution domain. This function has to be carefully hand ..."
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Many signal and image estimation problems such as maximum entropy reconstruction and positron emission tomography, require the minimization of a criterion containing a barrier function i.e., an unbounded function at the boundary of the feasible solution domain. This function has to be carefully handled in the optimization algorithm. When an iterative descent method is used for the minimization, a search along the line supported by the descent direction is usually performed at each iteration. However, standard line search strategies tend to be inefficient in this context. In this paper, we propose an original line search algorithm based on the majorizeminimize principle. A tangent majorant function is built to approximate a scalar criterion containing a barrier function. This leads to a simple line search ensuring the convergence of several classical descent optimization strategies, including the most classical variants of nonlinear conjugate gradient. The practical efficiency of the proposal scheme is illustrated by means of two examples of signal and image reconstruction. 1.
Duality Of Transformation Functions In The Interior Point Methods
, 1996
"... . In this paper a duality of transformation functions in the interior point method is treated. A dual pair of convex or linear programming problems is considered and the primal problem is transformed by the parametrized transformation function of a more general form than logarithmic is. The construc ..."
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. In this paper a duality of transformation functions in the interior point method is treated. A dual pair of convex or linear programming problems is considered and the primal problem is transformed by the parametrized transformation function of a more general form than logarithmic is. The construction of the parametrized transformation function for the dual problem is carried out so that both transformation functions were dual. The result obtained explains the unlucid construction of dual transformation functions so far known as a special case of a simple general principle of constructing dual transformation functions. 1. Introduction In the framework of the interior point methods (IPM) the linear programming problem (LP) min \Phi c T x j Ax = b; x 0 \Psi ; x; c 2 R n ; b 2 R m ; A 2 R m\Thetan is solved using logarithmic transformation function (1) T (x; ¯) = c T x \Gamma ¯ n X i=1 ln x i where x 2 P o = fx j Ax = b; x ? 0g and ¯ ? 0 is a parameter. The standard...
Computational Optimization and Applications manuscript No.
, 2009
"... (will be inserted by the editor) A MajorizeMinimize line search algorithm for barrier functions ..."
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(will be inserted by the editor) A MajorizeMinimize line search algorithm for barrier functions