Mathematical Sciences: Algorithms for Convex Programming-Interior Point and Proximal Point Methods

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This project will support research in the analysis and development of algorithms for large-scale convex programming problems. Two of the most versatile algorithms have been interior point methods and splitting methods based on the proximal point algorithm. Specific areas of research will include: (1) the interior point transformation, involving the analysis of the mapping in interior point methods that sends an interior feasible pair to its coordinatewise product; (2) infeasible interior point methods, including bridging the gap between the feasible and infeasible interior point methods; (3) convergence of the proximal point method, including the analysis of the global and local convergence behavior of the proximal point method and its modifications; and (4) interior point methods for nonlinear problems, involving the development of practical interior point methods, especially primal-dual methods, for special classes of nonlinear programs and monotone complementarity problems. Two of the most important modern algorithms for the numerical solution of large-scale convex programming problems have been interior point methods and proximal point methods. Both of these methods have had a major impact on modern numerical solution techniques for problems that were previously thought to be intractable because of their massive size.