Convex Analysis, Monotone Operator Theory and Algorithms

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Numerous problems in mathematics, engineering and the physical sciences may be modelled as a feasibility problem, which asks to find a solution satisfying various given constraints. A classical example is to find a solution to a system of linear equations. In many cases, the constraints allow us to compute the projection or nearest point mapping. Indeed, projection methods are popular iterative procedure to solve feasibility problems using the projections of the constraints. Projection methods have fascinated several outstanding researchers including John von Neumann, Norbert Wiener and Nobel prize winner Sir Godfrey Hounsfield who used these methods in the first Computer-Aided Tomography (CAT) scanner. Today, these methods continue to be of great interest in classical and modern areas such as crystallography and sparse feasibility, where the constraints are often non-convex. While a large body of results and algorithms exists, there are still many tantalizing important open problems including: 1) What can be said when the constraints are non-convex? 2) What happens if there is no solution (e.g., due to measurement errors)? 3) Can these methods be modified to work for more general optimization problems? 4) What happens if we apply projections that are based on non-Euclidean distances? I shall use modern tools from Convex Analysis, Monotone Operator Theory and Variational Analysis to advance the knowledge of projection methods and to make progress on the four open problem listed above. The skills my students and postdoctoral fellows will acquire will prepare them for successful careers in academia and industry which are critical to maintaining Canada's leadership in Science and Engineering.