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"Faynberg, Igor"
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The investigation and implementation of the power -series primal -dual algorithm for solving separable convex quadratic problems on the AT\\&T KORBX RTM processor
The seminal discovery of Narendra Karmarkar has led to rapid development of both the theory and practical implementation of new mathematical programming algorithms. In less than three years after the discovery, the Advanced Decision Support Systems Group of AT&T Bell Laboratories has developed the AT&T KORBX$\\circler$ System, which implements several variants of the Karmarkar algorithm on a powerful parallel/vector mini-supercomputer. This dissertation integrates advances in the algorithm and optimization theories with the state-of-the-art programming techniques. The main contribution of this dissertation is that it describes in detail all steps that have led to the implementation of a new Separable Quadratic Programming solver based on the Karmarkar-type algorithm in the context of the existing system. These steps include: (1) Investigation of the primal-dual path-following algorithm for solving large (general) Convex Quadratic Programs and application of the power-series acceleration technique to this algorithm. The resulting formulae are published for the first time; (2) Design of the new Mathematical Programming System Architecture and its integration with the existing one; (3) Implementation of the Separable Quadratic Program Solver on the AT&T KORBX System, a vector-concurrent mini-supercomputer; (4) Application of the visualization techniques to tuning the systems parameters. As the result, our optimizer is able to solve the Separable Quadratic Programming problems that could not be solved before. In addition, in this dissertation we systematically use the object-oriented approach and demonstrate how we are able to retain main structures and most of the code derived for the simplest case (e.g., the case of problems with only standard variables), to solve problems with upper-bounded, free, and artificial variables. We believe that some of our byproduct findings (such as the recursive property of the algorithm for solving problems with upper-bounded variables and the equivalence of the original and relaxed problems with free variables), are interesting on their own.
Dissertation