Nonlinear Optimization: Algorithms, Theory and Software

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Numerical optimization plays an essential role in a wide variety of scientific and engineering applications. Medical imaging, electrical power network simulations, computational finance, and atmospheric sciences make extensive use of optimization models to simulate real-life phenomena. As these models become increasingly more complex and incorporate a larger amount of data, the demands placed on optimization techniques have surpassed their capabilities. This proposal presents three projects designed to address this challenge.The first project proposes matrix-free methods for very large constrained optimization problems; we have in mind applications where the number of variables and constraints range in the millions. The new algorithms compute inexact Newton steps by applying iterative methods to the inner linear systems of equations. Questions to be addressed in this research include the appropriate management of inexactness, nonconvexity, and Jacobian singularity. The second project concerns the development of open-source software for problems in which the constraints are defined by the discretization of partial differential equations. The new optimization solvers will be created in collaboration with Argonne National Laboratory and will operate in a matrix-free environment. The third project investigates procedures for detecting if a nonlinear optimization problem is feasible. This question has not received sufficient attention in spite of its importance in mixed integer nonlinear programming and in parametric studies of optimization models. The goal is to develop new optimization techniques that transition smoothly between optimization and feasibility, and vice versa. The intellectual merits of the proposed activity lie in the complexity of designing optimization methods that are capable of dealing with nonconvexities and nonlinearities. The broader impacts resulting from this work will be seen in the successful application of the new algorithms and software in areas such as circuit simulation, computational chemistry, medical imaging, atmospheric sciences, and machine learning. The principles and ideas developed in this project will stimulate future research in new areas of application.