Structured Non-Smooth Optimization: Theory and Methods

研究人员开发理论和算法来解决当今商业，科学和工程应用中出现的优化问题，通常与数据分析问题有关，其目的是更好地为替代选择的决策提供信息。 该项目的主题出现在模型选择技术（用于社交网络的识别和选择最能告知健康结果的协变量等应用），成像应用，以及废物流和车辆的动态，跟踪和控制。 通常在应用中，目标函数（待优化的函数）可能是非光滑的、非凸的或高维的;这些特征中的每一个都对经典优化方法提出了重大挑战。 该项目的一个关键方面是使用非光滑技术，允许在所需的解决方案中引入或发现特殊属性，如稀疏性，稳定性和鲁棒性。 这项工作分为四个主要领域福尔斯。 第一个关注的BFGS方法和矩阵割线方法的研究非光滑凸优化（为什么BFGS的作品，以及它不清楚），和算法的设计，自动发现一个所谓的UV分解的目标函数。 这里U是光滑部分，V是非光滑部分。 这种分解对于快速解决方案识别是必不可少的，因为它们允许人们沿着具有陡峭侧边（V）的平滑谷（U）朝向最优解。 第二个检查的速度，获得的解决方案，以及其准确性，这取决于功能和参数的输入。 第三个检查的线性动力系统发生，例如，在轨道动力学或药物代谢的最佳稳定性和控制的确定。 这里的目标是帮助识别最佳解决方案。 事实上，在许多情况下，识别最优性的方法仍然是未知的。 最后一项任务涉及开发新的平滑方法，用于矩阵空间上的优化问题，例如社交网络发现中出现的问题。 这些问题通常是非常高维的，目标函数的非光滑性是发现数据中隐藏结构的关键。 这里的目标是开发光滑的近似，其解决方案可以快速计算，其接近真正的解决方案是精确控制。 研究生参与研究。更技术性的研究领域是（i）非光滑凸优化的BFGS和矩阵割线方法，（ii）凸复合优化的局部和全局收敛理论，（iii）非对称矩阵谱函数的变分分析，以及（iv）广义矩阵分式函数和矩阵空间上的光滑。 BFGS方法的研究集中在迭代与光滑、凸、一致逼近之间的关系。 凸组合问题的研究是分析广义方程罗宾逊方法的算法性能。 非对称矩阵的研究集中在将变分方法推广到困难的非负情况。 最后一个领域考虑了广泛的矩阵优化问题的嵌入在一个光滑的设置，通过使用infimal投影与广义矩阵分数函数引入的调查员和合作者。 该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The investigator develops theory and algorithms to solve optimization problems arising in present-day business, science, and engineering applications, often specifically related to data analysis problems where the aim is to better inform decisions about alternative choices. The project's topics arise in model selection techniques (used in applications such as the identification of social networks and the selection of covariates that best inform health outcomes), in imaging applications, and in the dynamics, tracking, and control of waste streams and of vehicles. Commonly in applications the objective function (the function to be optimized) may be nonsmooth, nonconvex, or high-dimensional; each of these features presents major challenges for classical optimization methods. A key aspect of the project is the use of nonsmooth techniques that allow for the introduction or discovery of special properties in the desired solution, such as sparsity, stability, and robustness. The work falls into four main areas. The first concerns study of the BFGS method and matrix secant methods for nonsmooth convex optimization (why BFGS works as well as it does is unclear), and the design of algorithms that automatically discover a so-called UV-decomposition of the objective function. Here the U is the smooth part and the V is the nonsmooth part. Such decompositions are essential for rapid solution identification as they allow one to follow a smooth valley (U) with steep sides (V) toward the optimal solution. The second examines the rate at which a solution is obtained as well as its accuracy, depending on functional and parameter inputs. The third examines the determination of optimal stability and control of linear dynamical systems occurring, for example, in orbital dynamics or drug metabolism. The goal here is to help recognize optimal solutions. Indeed, in many cases methods for identifying optimality remain unknown. The final task concerns the development of novel smoothing methodologies for optimization problems over matrix spaces, such as those occurring in social network discovery. These problems are typically very high-dimensional, and nonsmoothness of the objective function is the key to discovering hidden structures within the data. Here the goal is to develop smooth approximations whose solution can be rapidly computed and whose proximity to the true solution is precisely controlled. Graduate students participate in the research.More technically, the areas of study are (i) BFGS and matrix secant methods for nonsmooth convex optimization, (ii) local and global convergence theory for convex-composite optimization, (iii) variational analysis of spectral functions for non-symmetric matrices, and (iv) the generalized matrix fractional function and smoothing on matrix spaces. The study of BFGS methods focuses on the relationship between the iterates and smooth, convex, uniform approximations. The study of convex composite problems analyses the behavior of algorithms using Robinson's method of generalized equation. The study of nonsymmetric matrices focuses on extending variational techniques to the difficult nonderogatory case. The final area considers the embedding of a wide range of matrix optimization problems in a smooth setting by the use of infimal projection with the generalized matrix fractional function introduced by the investigator and collaborators. Graduate students participate in the research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.