Interior-point methods of optimization: extensions and applications

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Four topics are discussed in the proposal: explicit and approximate computations of universal barrier functions for various classes of semi-infinite programming problems, infinite-dimensional primal-dual algorithms with applications to multi-criteria and robust control problems, wireless communication and global optimization based on convex relaxations and a generalization of the theory of interior-point algorithms to Riemannian manifolds. Proposal introduces several new ideas and techniques for better undersanding the mathematical structure of interior-point algorithms. PI heavily relies upon classical results of M. Krein, A. Nudelman, I. Schoenberg on multidimensional versions of isoperimetric inequalities and totally positive matrices in computation of new important classes of universal barrier functions, on the theory of infinite-dimensional Jordan algebras for the analysis of infinite-dimensional primal-dual algorithms and control applications. PI brings some number-theoretic ideas related to Hilbert identities and classical constructions of so-called spherical designs to the global optimization. Semi-infinite programming problems appear in various applications: separation of sets in pattern recognition (e.g. medical diagnostics , identification with applications in home land security), environmental policies, robustness in Bayesian statistics, optimal experimental design in regression, the efficiency of industrial processes, filtering design in electrical engineering etc. The progress in resolving algorithmic issues for this class of problems may potentially have a tremendous impact on various applications (especially, in the situations where fast and optimal online decisions are necessary). Development of software for solving robust , multi-criteria control problems may be of importance in such diverse applications as stabilization of various complex structures in extreme situations (earthquakes, overloads of energy systems), prediction of the behavior of stock markets and control of spacecrafts.