Construction of Functions from Nonlinear Transformations (PhD Defense)

This thesis focuses on modeling, analyzing, and solving problems involving nonlinear equality constraints from the novel perspective of fixed point theory and monotone operator theory. It is shown that the class of nonlinearities involving firmly nonexpansive operators is broad enough to represent many equations arising in applications, even when the original equations feature non-Lipschitzian or even discontinuous operators. Adopting this model leads to feasibility and best approximation algorithms which are proven to converge to an exact solution of such equations from any initial point. Best approximation problems subjected to these nonlinear equations are solved with a new strongly convergent block-iterative algorithm that features an extrapolated relaxation scheme which exploits the presence of affine constraints. To address potentially inconsistent feasibility problems involving firmly nonexpansive equations, we propose a relaxation in the form of a variational inequality problem. Conditions for the existence of solutions to the relaxed problem are derived and a block-iterative algorithm is proposed for its solution. Next, block-iterative algorithms for fully nonsmooth convex minimization are analyzed, and their relative performance on concrete large-scale problems is assessed. Throughout, the theoretical and numerical aspects of this work are illustrated by applications to image processing, signal processing, and machine learning.