Structured convex optimization with applications

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The proposal is a response to a large demand on analytics, and optimization in particular, in Canada, and specifically in Alberta. It supports a goal to establish Calgary as a strong-hold in optimization by building an active industrial mathematics lab. The optimization lab is envisioned as a focal point, producing top quality research and a constant stream of employable HQP. Encompassing (1) applications, (2) theory and (3) numerical algorithms, optimization is ubiquitous to science and engineering. We pursue all three intertwined directions, but above all are motivated by the applications. (1) Applications: our central application area is radiation therapy (RT) in cancer treatment, where optimization plays a key role. In 2007 cancer surpassed cardiovascular disease as the leading cause of death. It is estimated that 2 out of 5 Canadians will develop cancer during their lifetimes; 1 out of 4 will die from cancer. In turn, over 50% of cancer patients are treated with RT. Thus, improving the RT efficacy is an important concern. Due to the large problem size setting 1,000s of parameters controlling radiation exposure modern RT planning is very challenging. Optimization methods are used to derive better treatment plans that deliver lethal dose to the tumor surrounded by healthy tissues to be spared. A main challenge in RT planning is incorporation of dose-volume requirements (DVR). DVR ensure the target coverage and survivability of healthy structures. Due their complexity, conventional models to include DVR are computationally intractable. Sadly, an over decade-old call to “optimization experts (to) improve our ability to solve these difficult problems” in prime applied math journal (Shepard et al, SIAM Review 41(4), 721744, 1999) is still open today. A recently discovered interplay between the dose moments and the DVR paves an innovative alternative route to solving this problem. A preliminary investigation shows that the approach offers an opportunity to optimize RT plans under DVR in near-real time. We aim to capitalize on this discovery and further investigate the above methods, targeting the development of a new RT optimization framework. In turn, this opens a road to a novel technology in RT, benefiting Canadians and cancer patients world-wide. (2,3) Theory and Algorithms: motivated by the above, we focus on structured convex optimization. Despite recent dramatic advances in optimization theory and computational capacities, we still fall short of solving many challenging real world problems such as RT planning. Better theory and implementations are desperately needed. To improve the theory, we probe into provable limits of IPM-family of most efficient optimization methods known to-date. On the implementation side, we target the development of modular IPM solver to enrich the class of problems that can be solved and enable computational enhancements offered by advanced IT such as GP-GPU.