Structured convex optimization with applications

该提案是对加拿大，特别是阿尔伯塔省对分析和优化的大量需求的回应。它通过建立一个活跃的工业数学实验室来支持将卡尔加里建立为优化据点的目标。优化实验室被设想为一个焦点，产生高质量的研究和源源不断的就业HQP。优化包括(1)应用，(2)理论和(3)数值算法，在科学和工程中无处不在。我们追求这三个相互交织的方向，但最重要的是由应用程序驱动。(1)应用领域：我们的核心应用领域是放射治疗（RT）在癌症治疗中的应用，其中优化起着关键作用。2007年，癌症超过心血管疾病，成为导致死亡的主要原因。据估计，五分之二的加拿大人一生中会患癌症；四分之一的人将死于癌症。反过来，超过50%的癌症患者接受了RT治疗，因此，提高RT疗效是一个重要的问题。由于问题规模大，需要设置上千个控制辐射暴露的参数，现代放射治疗规划非常具有挑战性。利用优化方法得出更好的治疗方案，将致死剂量传递给被健康组织包围的肿瘤。放疗计划的一个主要挑战是纳入剂量-体积要求（DVR）。DVR确保了健康建筑的目标覆盖率和生存能力。由于其复杂性，包含DVR的传统模型在计算上难以处理。可悲的是，一个十多年前的“优化专家”电话。（以）提高我们解决这些难题的能力”，主要应用数学期刊（Shepard et al, SIAM Review 41(4), 721-744, 1999）今天仍然开放。最近发现的剂量矩和DVR之间的相互作用为解决这一问题提供了一条创新的替代途径。初步研究表明，该方法为近实时地优化DVR下的RT计划提供了机会。我们的目标是利用这一发现，进一步研究上述方法，目标是开发一个新的RT优化框架。反过来，这为RT的新技术开辟了一条道路，使加拿大人和全世界的癌症患者受益。（2,3）理论与算法：受上述启发，我们重点研究结构化凸优化。尽管最近在优化理论和计算能力方面取得了巨大的进步，但我们仍然无法解决许多具有挑战性的现实世界问题，例如RT规划。我们迫切需要更好的理论和实现。为了完善理论，我们探讨了迄今为止已知的最有效的优化方法的ipm族的可证明极限。在实现方面，我们的目标是开发模块化IPM求解器，以丰富可解决的问题类别，并使GP-GPU等先进IT提供的计算增强成为可能。

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), 721-744, 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.