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

我的研究旨在应用连续优化松弛和利用结构，以获得有效和准确地解决大规模困难的数值问题的实用算法。这些问题出现在数据科学中，通常是非凸和棘手的。一个众所周知的应用是低秩矩阵完成（LRMC）问题，例如网络电影评级问题，即，可以填充缺失的条目以便向客户做出好的推荐。其他应用包括：模拟调度和定位问题的硬离散优化问题，例如二次分配问题;鲁棒主成分分析，包括从高度损坏的监视数据中恢复测量值;分子构象和蛋白质折叠传感器网络定位以及多项式方程的真实的解。 在整个过程中，我强调具体的应用程序与广泛的数值tests.Many当前的优化建模和凸松弛硬非凸问题通常会导致严格的可行性损失，内部的可行集。这可能导致理论和数值上的困难。而不是一个缺点，我们把这种规律性的损失变成一个很大的优势，使用一种技术称为面部缩小（FR）。特别是，这导致了非常有效的算法，许多棘手的问题。其结果是，我们可以解决巨大的问题，往往不需要一个数值优化求解器。对于矩阵完成类型的问题，在无噪声的情况下，得到非常高的精度。在有噪声的情况下，可以使用暴露面向量的概念来避免舍入误差的积累，并获得噪声量级内的解。此外，FR技术似乎完全适合一阶方法，因为可以保持两组变量并在它们之间交替投影。 我所做的工作在计算机科学中的数据科学和机器学习领域以及运筹学中有许多应用。特别地，规律性、严格可行性的丧失出现在如上所述的令人惊讶的许多应用中。FR技术允许比目前处理的更大问题的更精确的解决方案。我们的方法的新奇在于，我们可以利用问题的结构来识别规律性的损失，然后利用这种损失来获得一个既稳定又减小大小的重新表述。我们强调FR作为一种预处理技术，并显示其数学优雅，几何透明性和计算潜力。

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.