Convex Analysis, Monotone Operator Theory and Algorithms

数学、工程和物理科学中的许多问题可以建模为可行性问题，它要求找到满足各种给定约束的解决方案。一个经典的例子是求线性方程组的解。在许多情况下，约束允许我们计算投影或最近点映射。实际上，投影方法是使用约束的投影来解决可行性问题的流行迭代过程。投影方法吸引了一些杰出的研究人员，包括约翰·冯·诺依曼，诺伯特·维纳和诺贝尔奖赢家戈弗雷·亨斯菲尔德爵士，他在第一台计算机辅助断层扫描（CAT）扫描仪中使用了这些方法。今天，这些方法仍然在经典和现代领域，如晶体学和稀疏可行性，其中的约束往往是非凸的极大兴趣。虽然有大量的结果和算法存在，仍然有许多诱人的重要的开放问题，包括：1）什么可以说，当约束是非凸的？2)如果没有解决方案会发生什么（例如，由于测量误差）。3)这些方法是否可以修改以适用于更一般的优化问题？4)如果我们应用基于非欧几里德距离的投影会发生什么？我将使用现代工具，从凸分析，单调算子理论和变分分析，以提高知识的投影方法，并取得进展的四个开放的问题上面列出。我的学生和博士后研究员将获得的技能将为他们在学术界和工业界的成功职业生涯做好准备，这对保持加拿大在科学和工程领域的领导地位至关重要。

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.