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On the Classical Logarithmic Barrier Function Method for a Class of Smooth Convex Programming Problems
, 1990
"... In this paper we propose a largestep analytic center method for smooth convex programming. The method is a natural implementation of the classical method of centers. It is assumed that the objective and constraint functions fulfil the socalled Relative Lipschitz Condition, with Lipschitz constant ..."
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Cited by 14 (4 self)
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In this paper we propose a largestep analytic center method for smooth convex programming. The method is a natural implementation of the classical method of centers. It is assumed that the objective and constraint functions fulfil the socalled Relative Lipschitz Condition, with Lipschitz constant M ? 0. A great advantage of the method, above the existing pathfollowing methods, is that the steps can be made long by performing linesearches. In our method we do linesearches along the Newton direction with respect to a strictly convex potential function if we are far away from the central path. If we are sufficiently close to this path we update a lower bound for the optimal value. We prove that the number of iterations required by the algorithm to converge to an ffloptimal solution is O((1 +M 2 ) p nj ln fflj) or O((1 +M 2 )nj ln fflj), dependent on the updating scheme for the lower bound.
Convergence property of the IriImai algorithm for some smooth convex programming problems
 Journal of Optimization Theory and Applications
, 1994
"... In this paper, the IriImai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions sat ..."
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Cited by 7 (5 self)
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In this paper, the IriImai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity (called the harmonic convexity in this paper). A characterization of this convexity condition is given. In Ref. 14, the same convexity condition is used to prove the convergence of a pathfollowing algorithm. The IriImai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior point algorithms for smooth convex programming are mainly based on the pathfollowing approach.
Inverse Barrier Methods for Linear Programming
 REVUE RAIROOPERATIONS RESEARCH
, 1991
"... In the recent interior point methods for linear programming much attention has been given to the logarithmic barrier method. In this paper we will analyse the class of inverse barrier methods for linear programming, in which the barrier is P x \Gammar i , where r ? 0 is the rank of the barrier. ..."
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Cited by 6 (1 self)
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In the recent interior point methods for linear programming much attention has been given to the logarithmic barrier method. In this paper we will analyse the class of inverse barrier methods for linear programming, in which the barrier is P x \Gammar i , where r ? 0 is the rank of the barrier. There are many similarities with the logarithmic barrier method. The minima of an inverse barrier function for different values of the barrier parameter define a 'central path' dependent on r, called the rpath of the problem. For r # 0 this path coincides with the central path determined by the logarithmic barrier function. We introduce a metric to measure the distance of a feasible point to a point on the path. We prove that in a certain region around a point on the path the Newton process converges quadratically. Moreover, outside this region, taking a step into the Newton direction decreases the barrier function value at least with a constant. We will derive upper bounds for the total ...