Splitting methods in optimization: Beyond consistency and convexity.

优化问题出现在科学、工程和医疗保健的许多领域，要求找到受约束函数的最小值或最大值。分裂算法是一种强大的优化方法，旨在解决在实践中经常遇到的约束凸（但不一定是光滑）优化问题。这些方法的例子是Douglas-Rachford法，正向-反向法和乘法器的交替方向法（ADMM）。具体的应用领域包括图像处理（例如，图像去噪问题），数据科学（例如，支持向量机问题），统计学和机器学习（例如，最小绝对收缩和选择算子（LASSO）问题）和金融（例如，投资组合优化）。寻找保证这些算法收敛到一个解的条件是一个非常活跃的研究领域。典型的收敛结果要求目标函数和/或约束的凸性，以及解的存在性。我的研究计划的长期目标是研究在没有最小化和/或凸性的情况下拆分方法的行为。调查这种情况的重要性源于这样一个事实，即它在实践中发生，但尚未得到很好的理解。我研究的这些方面与相位检索、强度调制放射治疗和深度学习（通过流行的神经网络的非凸损失函数）有关。我的研究计划的短期目标总结如下。1. 使用来自凸和变分分析的工具，我将探索在不一致和/或非凸情况下不同分裂算法的分析行为。其中一些方法在应用于这些情况时显示出很大的希望；然而，目前还缺乏深入的分析。2. 在这些分析结果的基础上，我将利用动力系统和非线性算子理论的技术来开发分析的算法对应物。在这种情况下，我们必须处理缺乏收敛证明中通常使用的性质的算子（例如，非凸集上的投影）。3. 我的研究计划的第三个组成部分是计算对应物。本研究包括数值测试和与其他现有方法的计算成本比较，以及开发软件包。影响：一方面，我们取得的进展将对采用分裂方法的科学家以及工业和医疗保健部门产生重大兴趣，因为用于模拟工业问题的函数通常是非凸的。另一方面，我的学生和博士后将获得强大的分析和计算基础。这将有助于优化理论和相关软件的发展。这两个方面都将对加拿大在科学和技术方面的领导作用作出重大贡献。

Optimization problems, which appear in many fields of science, engineering and healthcare, ask to find minimizers or maximizers of functions subject to constraints. Splitting algorithms are powerful optimization methods which aim to solve constrained convex (but not necessarily smooth) optimization problems, which are oftentimes encountered in practice. Examples of these methods are the Douglas-Rachford method, the forward-backward method and the alternating direction method of multipliers (ADMM). Concrete areas of applications include image processing (e.g., image denoising problems), data science (e.g., support vector machine problems), statistics and machine learning (e.g., the least absolute shrinkage and selection operator (LASSO) problem) and finance (e.g., portfolio optimization). Finding conditions that guarantee the convergence of these algorithms to a solution is a very active area of research. Typical convergence results require the convexity of the objective function and/or the constraints, as well as the existence of solutions. The long-term objective of my research program is to investigate the behaviour of splitting methods in the absence of minimizers and/or convexity. The importance of investigating this situation stems from the fact that it occurs in practice but is not well understood. These aspects of my research connect to phase retrieval, intensity-modulated radiation therapy, and deep learning (via the popular but nonconvex loss function from neural networks). The short-term objectives of my research program are summarized below. 1. Using tools from convex and variational analysis, I will explore the analytic behaviour of different splitting algorithms in the inconsistent and/or nonconvex case. Some of these methods have shown great promise when applied to these cases; however, an in-depth analysis is still lacking. 2. Building on these analytical results, I will utilize techniques from dynamical systems and nonlinear operator theory to develop the algorithmic counterpart of the analysis. In this case, we must deal with operators (for instance, projections onto nonconvex sets) that lack properties typically utilized in convergence proofs. 3. The third component of my research program is the computational counterpart. This research includes numerical testing and comparison of the computational cost of these methods to other existing ones, as well as developing software packages. Impact: On the one hand, the progress we make will be of significant interest for scientists employing splitting methods as well as the industrial and healthcare sectors, since the functions used to model industrial problems are often nonconvex. On the other hand, my students and postdoctoral fellows will acquire a strong analytic and computational foundation. This will contribute to developments in optimization theory and related software. Both aspects will significantly contribute to Canada's leading role in science and technology.