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21
A Polynomial Method of Weighted Centers for Convex Quadratic Programming
 JOURNAL OF INFORMATION & OPTIMIZATION SCIENCES
, 1991
"... A generalization of the weighted central pathfollowing method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central pathfollowing method for linear programming and the interior point methods for convex quadratic programming. B ..."
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Cited by 3 (2 self)
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A generalization of the weighted central pathfollowing method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central pathfollowing method for linear programming and the interior point methods for convex quadratic programming. By means of the linear approximation of the weighted logarithmic barrier function and weighted inscribed ellipsoids, `weighted' trajectories are defined. Each strictly feasible primal dual point pair define such a weighted trajectory. The algorithm can start in any strictly feasible primaldual point pair that defines a weighted trajectory, which is followed through the algorithm. This algorithm has the nice feature, that it is not necessary to start the algorithm close to the central path and so additional transformations are not needed. In return, the theoretical complexity of our algorithm is dependent on the position of the starting point. Polynomiality is proved under the usual mild conditions.
A Potential Reduction Method for a Class of Smooth Convex Programming Problems
, 1990
"... In this paper we propose a potential reduction method for smooth convex programming. It is assumed that the objective and constraint functions fulfil the socalled Relative Lipschitz Condition, with Lipschitz constant M ? 0. The great advantage of this method, above the existing pathfollowing metho ..."
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In this paper we propose a potential reduction method for smooth convex programming. It is assumed that the objective and constraint functions fulfil the socalled Relative Lipschitz Condition, with Lipschitz constant M ? 0. The great advantage of this method, above the existing pathfollowing methods, is that it allows 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.
A potential reduction variant of Renegar's shortstep pathfollowing method for linear programming
, 1990
"... In this paper we propose a new polynomial potential reduction method for linear programming, which can also be seen as a largestep pathfollowing method. In our method we do an (approximate) linesearch along the Newton direction with respect to Renegar's strictly convex potential function if ..."
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Cited by 2 (2 self)
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In this paper we propose a new polynomial potential reduction method for linear programming, which can also be seen as a largestep pathfollowing method. In our method we do an (approximate) linesearch along the Newton direction with respect to Renegar's strictly convex potential function if the iterate is far away from the central trajectory. If the iterate lies close to the trajectory we update the lower bound for the optimal value. Dependent on this updating scheme the iteration bound can be proved to be O( p nL) or O(nL). Our method differs from the recently published potential reduction methods in the choice of the potential function and the search direction.
A Knowledge Representation for ConstraintSatisfaction Problems
, 1990
"... AbstractIn this paper we present a general representation for constraint satisfaction problems (CSP) and a framework for reasoning about their solution that unlike most constraintbased relaxation algorithms. stresses the need for a "natural " encoding of constraint knowledge and ..."
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AbstractIn this paper we present a general representation for constraint satisfaction problems (CSP) and a framework for reasoning about their solution that unlike most constraintbased relaxation algorithms. stresses the need for a &quot;natural &quot; encoding of constraint knowledge and can facilitate making inferences for propagation, backtracking, and explanation. The representation consists oi two componenrs: a generateandtest problem solver which cclntains information about the problem variables, and a constraintdriven reasoner that manages a set of constraints, specified as arbitrarily complex Boolean expressions and represented in the form of a constraint network. This constraint network: incorporates control information (reflected in the syntax of the constraints) that is used for constaint propagaticn: contains dependency information that can be used for explanation and for dependencydirected backtracking; and is incremental in the sense that if the problem specification is modified, a new solution can be derived by modifying the existing solution. 1.
NP and Mathematics  a computational complexity perspective
 Proc. of the ICM 06
"... “P versus N P – a gift to mathematics from Computer Science” ..."
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“P versus N P – a gift to mathematics from Computer Science”
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"... I wish to thank my supervisor, Dr. Gautam Mitra, for the excellent guidance he has given me over the past four years. His encouragement and enthusiasm have greatly helped to maintain my motivation and interest in mathematical programming. The advisory staff of BruneI's computer unit, especially ..."
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I wish to thank my supervisor, Dr. Gautam Mitra, for the excellent guidance he has given me over the past four years. His encouragement and enthusiasm have greatly helped to maintain my motivation and interest in mathematical programming. The advisory staff of BruneI's computer unit, especially Mr R Pank, have been very helpful in overcoming computer problems. I wish to thank them for their expertise and willingness to help. Mrs Mary Storey and Mrs Pam Denham typed part of this thesis, and Mrs Barbara Yates drafted some of the diagrams, to all of these I am extremely grateful.
Technical Report Linear Programming 1
, 1997
"... Dedicated to George Dantzig on this the 50 th ..."