Exploiting Structure and Hidden Convexity in Hard, Large Scale Numerical Optimization

在困难的大规模数值优化中利用结构和隐藏凸性

• 批准号: RGPIN-2018-04028
• 负责人: Wolkowicz, Henry
• 金额: $ 4.01万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712115
• 关键词: Exploiting; Structure; Hidden; Convexity; Hard

My research aims to apply continuous optimization relaxations and to exploit structure in order to derive practical algorithms that efficiently and accurately solve large scale hard numerical problems. These problems arise in data science and are generally non-convex and intractable. One well known application is the low-rank matrix completion (LRMC) problem, such as the netflix movie ratings problem, i.e., one can fill missing entries in order to make good recommendations to customers. Other applications include: hard discrete optimization problems that model scheduling and location problems such as the quadratic assignment problem; robust principal component analysis that includes recovery of measurements from highly corrupted surveillance data; molecular conformation and protein folding sensor network localization and real solutions of polynomial equations. Throughout I emphasize specific applications with extensive numerical tests. Many current optimization modelling and convex relaxations for hard non-convex problems typically result in loss of strict feasibility, interior, of the feasible set. This can result in both theoretical and numerical difficulties. Rather than being a disadvantage, we turn this loss of regularity to a great advantage using a technique called facial reduction (FR). In particular, this has resulted in extremely efficient algorithms for many intractable problems. The result is that we can solve huge problems often without the need of a numerical optimization solver. For matrix completion type problems, in the noiseless case one gets extremely high accuracy. In the noisy case, one can use the notion of exposing vectors of the faces to avoid build up of round-off error and obtain solutions within the order of the noise. In addition, the FR technique appears to fit perfectly with first order methods as one can maintain two sets of variables and alternate projections between them. The work I am doing has many applications to the data science and machine learning areas in computer science and to operations research. In particular, the loss of regularity, strict feasibility, appears in surprisingly many applications as mentioned above. The FR techniques allow for more accurate solutions of larger problems than have currently been handled. The novelty of our approach is we can exploit the structures of problems to identify loss of regularity and then use this loss to our advantage to obtain a reformulation that is both stable and reduces the size. We emphasize FR as a preprocessing technique and display its mathematical elegance, geometric transparency and computational potential.

我的研究旨在应用连续优化松弛和开发结构，以获得有效和准确地解决大规模数值难题的实用算法。这些问题出现在数据科学中，通常是非凸的和棘手的。一个众所周知的应用是低秩矩阵补全（LRMC）问题，例如netflix电影评级问题，也就是说，一个人可以填补缺失的条目，以便向客户做出好的推荐。其他应用包括：模拟调度和位置问题的硬离散优化问题，如二次分配问题；强大的主成分分析，包括从高度损坏的监测数据中恢复测量；分子构象和蛋白质折叠传感器网络的定位和多项式方程的实解。在整个过程中，我强调具体的应用与广泛的数值测试。