Funding from the Air Force has played an instrumental role in my research. The broad goals of the proposed work were two-fold 1 to develop facial reduction algorithms for large-scale highly structured problems and 2 to develop new algorithms for nonsmooth and nonconvex problems, which converge rapidly under favorable conditions. In parallel, this project aimed to apply the techniques to pervasive large scale problems in computational mathematics and the applied sciences. All objectives were successfully met, as I explain below. The results of the research were summarized in over 20 publications in high calibre journals, including Math Prog., SIAM J. Optim., Math of Oper. Res. and Found. Comput. Math. Support from the AFOSR has funded three of my students, Kellie MacPhee, Scott Roy, and Courtney Paquette. The three students participated in all aspects of the project, coauthoring papers and presenting at conferences. Sections 1-5 describe the research highlights of the project, organized by topic the names of the three students appear in blue in the bibliographic citations. The final Section, 6, lists the conferences and colloquia that the PI attended as an invited speaker during the award period.