In this paper, we propose a line-search procedure for the logarithmic barrier function in the context of an interior point algorithm for convex quadratic programming. Preliminary testing shows that the proposed proced...
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In this paper, we propose a line-search procedure for the logarithmic barrier function in the context of an interior point algorithm for convex quadratic programming. Preliminary testing shows that the proposed procedure is superior to some other line-search methods developed specifically for the logarithmic barrier function in the literature.
In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called r...
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In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constant M > 0. In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an epsilon-optimal solution is O((1 + M2) square-root n\log-epsilon\) or O((1 + M2)n\log-epsilon\), depending on the updating scheme for the lower bound.
We provide a simple interpretation of the use of the logarithmic barrier function in convex and linear programming. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 90C05;90C25.
We provide a simple interpretation of the use of the logarithmic barrier function in convex and linear programming. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 90C05;90C25.
An interior point method for quadratically constrained convex quadratic programming is presented that is based on a logarithmic barrier function approach and terminates at a required accuracy of an approximate solutio...
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An interior point method for quadratically constrained convex quadratic programming is presented that is based on a logarithmic barrier function approach and terminates at a required accuracy of an approximate solution in polynomial time. This approach generates a sequence of unconstrained optimization problems, each of which is approximately solved by taking a single step in a Newton direction.
A new algorithm is presented using a logarithmic barrier function decomposition for the solution of the large-scale multicommodity network flow problem. Placing the com- plicating joint capacity constraints of the mul...
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A new algorithm is presented using a logarithmic barrier function decomposition for the solution of the large-scale multicommodity network flow problem. Placing the com- plicating joint capacity constraints of the multicommodity network flow problem into a logarithmicbarrier term of the objective function creates a nonlinear mathematical program with linear network flow constraints. Using the technique of restricted simplicial decomposition, we generate a sequence of extreme points by solving independent pure network problems for each commodity in a linear subproblem and optimize a nonlinear master problem over the convex hull of a fixed number of retained extreme points and the previous master problem solution. Computational results on a network with 3,300 nodes and 10,400 arcs are reported for four, ten and 100 commodities.
We give several reSSPIsults, some new and some old, but apparently overlooked, that provide useful characterizations of barrierfunctions and their relationship to problem function properties. In particular, we show t...
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We examine the sequence of local minimizers of the log-barrierfunction for a nonlinear program near a solution at which second-order sufficient conditions and the Mangasarian-Fromovitz constraint qualification are sa...
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We examine the sequence of local minimizers of the log-barrierfunction for a nonlinear program near a solution at which second-order sufficient conditions and the Mangasarian-Fromovitz constraint qualification are satisfied, but the active constraint gradients are not necessarily linearly independent, When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrierfunction in the vicinity of the nonlinear program solution, and we obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrierfunction and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrierfunction to the nonlinear programming solution.
We analyze robust stability properties of linear model predictive control schemes that are based on relaxed logarithmic barrier functions. It is shown that the corresponding closedloop system is globally exponentially...
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In this paper, the compensator based reduced order control design framework of Burns and King (J. Vibrations and Control, vol. 4, pp. 297-323, 1998) is modified to yield low order systems with guaranteed stability mar...
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In this paper, the compensator based reduced order control design framework of Burns and King (J. Vibrations and Control, vol. 4, pp. 297-323, 1998) is modified to yield low order systems with guaranteed stability margins. This result is achieved through use of a logarithmic barrier function. In addition, a reduced basis method is formulated in which the compensator equations are approximated on uneven grids;guaranteed stability margins are also included. The methods are tested numerically on a one dimensional, nonlinear, damped, hyperbolic structural control problem. Examples are provided to illustrate differences between systems designed by both reduced basis methods.
We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions ...
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We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmicbarrier families of problems, we show that these trajectories tend to an optimal solution which in general depends on the starting point. By considering the trajectories that arise from the Lagrangian multipliers of the above mentioned logarithmicbarrier families of problems, we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called 'centered' optimal solution of the dual problem. We also present results related to asymptotic direction of the affine scaling trajectories. We briefly discuss how to apply our results to linear programs formulated in formats different from the standard form. Finally, we extend the results to the primal-dual affine scaling algorithm.
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