9401871 Teboulle The research program will focus on proximal-like methods and their applications to nonlinear constrained optimization problems. Central to this study is a recently proposed new class of proximal-like mappings associated with closed convex functions, where the usual quadratic term is replaced by a kind of quasi-distance. When applied to the dual of an optimization problem, these give rise to wide variety of modified Lagrangians methods which have many attractive properties not present in the classical quadratic augmented Lagrangian. The research project will attempt to provide a unifying framework for the design and analysis of modified multiplier methods for solving nonlinear programming problems. The program's objective propose to sharpen and extend the mathematical theory needed to analyze the above algorithms. In particular, the research will consider new variants of modified Lagrangians for both convex and nonconvex programs, and will develop a complete convergence theory. Related decomposition algorithms within the proximal framework will also be investigated. This project will focus on the study and development of new methods for solving nonlinear constrained optimization problems. Such problems consist of minimizing/maximizing an objective subject to some constraints which are determined by the nature of the particular model/problem. Optimization techniques play a central role in the solution of fundamental problems arising in a wide range of applications such as: computerized tomography, finance, geophysical sciences, production and manufacturing, robotics, to mention just a few. One area of this research will concentrate on developing the mathematical theory needed to analyze a new class of methods for solving nonlinear problems. In particular, the PI will focus on the design and analysis of reliable and fast algorithms. The second main research area will concentrate on decomposition methods for large scale problems. Recen t advances in computer technology have motivated the search for constructing new algorithms for specific classes of large scale problems, by taking full advantage of the structure of this problem classes. Such problems arise for example in structural optimization, water distribution systems, and planning problems under uncertainty. Here, the PI will study and develop splitting methods, which are leading to highly parallel algorithms. In both areas, the role of optimization methods is central. Improvement of methods and their practical applications would thus have a beneficial impact on contributing to the solution of real life problems.
