Splitting methods in optimization: Beyond consistency and convexity.

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***Optimization problems, which appear in many fields of science, engineering and healthcare, ask to find minimizers or maximizers of functions subject to constraints. Splitting algorithms are powerful optimization methods which aim to solve constrained convex (but not necessarily smooth) optimization problems, which are oftentimes encountered in practice. Examples of these methods are the DouglasRachford method, the forward-backward method and the alternating direction method of multipliers (ADMM). Concrete areas of applications include image processing (e.g., image denoising problems), data science (e.g., support vector machine problems), statistics and machine learning (e.g., the least absolute shrinkage and selection operator (LASSO) problem) and finance (e.g., portfolio optimization).******Finding conditions that guarantee the convergence of these algorithms to a solution is a very active area of research. Typical convergence results require the convexity of the objective function and/or the constraints, as well as the existence of solutions. The long-term objective of my research program is to investigate the behaviour of splitting methods in the absence of minimizers and/or convexity. The importance of investigating this situation stems from the fact that it occurs in practice but is not well understood. These aspects of my research connect to phase retrieval, intensity-modulated radiation therapy, and deep learning (via the popular but nonconvex loss function from neural networks).******The short-term objectives of my research program are summarized below.******1. Using tools from convex and variational analysis, I will explore the analytic behaviour of different splitting algorithms in the inconsistent and/or nonconvex case. Some of these methods have shown great promise when applied to these cases; however, an in-depth analysis is still lacking. ***2. Building on these analytical results, I will utilize techniques from dynamical systems and nonlinear operator theory to develop the algorithmic counterpart of the analysis. In this case, we must deal with operators (for instance, projections onto nonconvex sets) that lack properties typically utilized in convergence proofs.***3. The third component of my research program is the computational counterpart. This research includes numerical testing and comparison of the computational cost of these methods to other existing ones, as well as developing software packages.******Impact: On the one hand, the progress we make will be of significant interest for scientists employing splitting methods as well as the industrial and healthcare sectors, since the functions used to model industrial problems are often nonconvex. On the other hand, my students and postdoctoral fellows will acquire a strong analytic and computational foundation. This will contribute to developments in optimization theory and related software. Both aspects will significantly contribute to Canada's leading role in science and technology.***