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

我的研究目的是应用连续优化松弛和开发结构，以得出有效和准确地解决大规模困难数值问题的实用算法。这些问题出现在数据科学中，通常是非凸的和难以解决的。一个众所周知的应用是低阶矩阵补全(LRMC)问题，例如Netflix电影评级问题，即，人们可以填充缺失的条目以向客户做出好的推荐。其他应用包括：模拟调度和选址问题的硬离散优化问题，如二次分配问题；稳健的主成分分析，包括从高度污染的监视数据恢复测量；分子构象和蛋白质折叠传感器网络定位和多项式方程的实解。在整个过程中，我通过大量的数值试验来强调具体的应用。*许多当前的优化建模和对硬非凸问题的凸松弛通常会导致丧失可行集的严格可行性。这可能会导致理论和数值上的困难。我们使用一种名为面部缩小(FR)的技术，将这种规则的丧失转化为巨大的优势，而不是一种缺点。特别是，这为许多棘手的问题带来了极其高效的算法。其结果是，我们可以解决巨大的问题，通常不需要数值优化求解器。对于矩阵补全类型的问题，在无噪音的情况下，可以获得极高的精度。在有噪声的情况下，可以使用曝光人脸向量的概念来避免舍入误差的累积，并在噪声的量级内获得解。此外，FR技术似乎非常适合一阶方法，因为人们可以保持两组变量和它们之间的交替投影。*我正在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领域都有很多应用。特别是，失去了规则性，严格的可行性，出现在上面提到的令人惊讶的许多应用中。FR技术允许对更大问题的更准确的解决方案，而不是目前处理的问题。我们方法的新奇之处在于，我们可以利用问题的结构来识别正则性损失，然后利用这种损失来获得既稳定又减小规模的重新公式。我们强调FR作为一种预处理技术，并展示了它的数学优雅、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.