In this work we consider a region R in R(n) given by a finite number of linear inequalities and having nonempty interior. We assume a point x degrees is given, which is close in certain norm to the analytic center of ...
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In this work we consider a region R in R(n) given by a finite number of linear inequalities and having nonempty interior. We assume a point x degrees is given, which is close in certain norm to the analytic center of R, and that a new linear inequality is added to those defining R. It is constructively shown how to obtain a perturbation of the right-hand side of this inequality such that the point x degrees is still close, in the same norm, to the analytic center of this perturbed polytope. This fact plays a central role in interior point postoptimality techniques for linear programming involving methods of centers.
A computational study of some logarithmicbarrier decomposition algorithms for semi-infinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmicbarrier ...
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The paper presents a logarithmicbarrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear a...
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The paper presents a logarithmicbarrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear program's optimal point. Other methods, like the ''central cutting'' plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmicbarrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.
Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier ...
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Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when the barrier parameter is reduced at a reasonable rate, as in a practical (long-step) method. Local analysis is presented indicating why a pure Newton step in a typical long-step barrier method for nonlinearly constrained optimization may be seriously infeasible, even when taken from an apparently favorable point;hence accurate calculation of the Newton direction does not guarantee an effective algorithm. The features described are illustrated numerically and connected to known theoretical results for well-behaved convex problems satisfying common assumptions such as self-concordancy. The contrasting nature of an approximate step to the desired minimizer of the barrierfunction is also discussed.
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.
In this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds fo...
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In this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely O(square-root nL) iterations for the short-step variant and O(nL) for the long-step variant. In the analysis of both variants, we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long-step method depends strongly on the fact that the usual search directions form a descent direction for the so-called primal-dual logarithmic barrier function.
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.
In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along proj...
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In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations is O(square-root nL) or O(nL), depending on how the barrier parameter is updated.
We present a path-following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour. This is done both for the problem in standard form and for its dual pr...
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We present a path-following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour. This is done both for the problem in standard form and for its dual problem. We also discuss some implementation strategies.
We propose a strategy for building up the linear program while using a logarithmicbarrier method. The method starts with a (small) subset of the dual constraints, and follows the corresponding central path until the ...
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We propose a strategy for building up the linear program while using a logarithmicbarrier method. The method starts with a (small) subset of the dual constraints, and follows the corresponding central path until the iterate is close to (or violates) one of the constraints, which is in turn added to the current system. This process is repeated until an optimal solution is reached. If a constraint is added to the current system, the central path will, of course, change. We analyze the effect on the barrierfunction value if a constraint is added. More importantly, we give an upper bound for the number of iterations needed to return to the new path. We prove that in the worst case the complexity is the same as that of the standard logarithmicbarrier method. In practice this build-up scheme is likely to save a great deal of computation.
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