Investigations in Interior Point Methods and Convex Programming

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AbstractThe goal of this project is to develop algorithms and tools for interior point methods. The investigator will continue his ongoing research into interior point methods with the aim of advancing several outstanding topics in semidefinite programming (SDP) and related problems. A better understanding of neighborhoods and paths is needed in SDP, and in programming over a symmetric cone in general. This is necessary in order to develop more efficient algorithms in terms of faster asymptotic convergence and numerical stability. The investigator intends to develop effective mathematical tools to deal with these issues. Programming over homogeneous cones and hyperbolic cones are likely to become the next emerging fields in interior point methods, and there is a need to develop efficient, long-step interior algorithms for such problems. Eventually, interior point methods will need to deal with even more complicated industrial applications which must be solved efficiently. The project will involve all of these areas of interior point methods; efficient algorithms will be devised in all cases, and the needed mathematical tools will be developed. The project also has two other goals. The first is an extension of duality theory beyond its traditional convexity, and the second one is the development of faster proximal point algorithms for convex programming. Interior point methods have been successful in solving many large scale industrial problems in industry: in civil and electrical engineering, management, communication networks, finance, and others. Further applicability of these methods depends on a better understanding of their behavior and continuous development of appropriate software. This project aims to search for efficient algorithms and improved mathematical tools so that large scale optimization problems arising from diverse industrial disciplines can be efficiently solved.