Convex Analysis, Monotone Operator Theory and Algorithms

凸分析、单调算子理论与算法

• 批准号: 216877-2013
• 负责人: Bauschke, Heinz
• 金额: $ 2.11万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635673
• 关键词: Convex; Analysis; Monotone; Operator; Theory

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.

数学、工程和物理科学中的许多问题都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，投影法是利用约束的投影来求解可行性问题的一种流行的迭代方法。投影方法吸引了几位杰出的研究人员，包括约翰·冯·诺伊曼、诺伯特·维纳和诺贝尔奖得主戈弗雷·霍斯菲尔德爵士，他们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，这些方法继续在经典和现代领域引起极大的兴趣，如晶体学和稀疏可行性，其中约束通常是非凸的。虽然存在大量的结果和算法，但仍然存在许多诱人的重要开放问题，包括：1)当约束是非凸时，我们可以说什么？2)如果没有解决方案（例如，由于测量误差）怎么办？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得距离的投影会发生什么？我将使用凸分析、单调算子理论和变分分析等现代工具来推进投影方法的知识，并在上面列出的四个开放问题上取得进展。我的学生和博士后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，这对于保持加拿大在科学和工程领域的领导地位至关重要。