Theory and Applications of Stochastic First-order Methods for Large-scale Stochastic Convex Optimization

这项拨款为研究解决随机和大规模凸规划（CP）问题的有效算法提供资金。CP算法的发展面临着一些新的挑战。首先，不确定性的结合，以及可能的动态，大大增加了求解CP的难度。其次，许多新兴应用（如统计学习）产生的CP问题的规模远远超过了现有多项式时间求解器的能力。在这个项目中，我们将研究一类新的和有前途的求解随机CP的随机一阶方法，并建立它们的收敛性和大偏差性。此外，利用这些随机方法作为驱动力，我们的目标是通过开发新的随机化方案来提高我们求解超大规模CP的能力。最后，我们将提出各种统计学习模型的优化求解器，这些模型对于在信息时代将数据转化为知识至关重要。如果成功，这项研究的结果将导致算法创新，以解决物流、金融、能源、医药、国防和科学等各个领域出现的大规模和随机CP问题。在绿色能源系统中，随机规划模型用于水火发电系统的长期运行规划，以处理随机流。在医学上，统计学习技术可以应用于诊断乳腺癌，通过利用单个细胞的特征来区分乳腺肿块的良恶性。这些只是本研究将要研究的影响社会的SP、大规模CP和统计学习问题的几个应用实例。此外，本研究将通过开发cp求解器软件，为开放式研究基础设施的优化做出贡献。我们也会让学生参与我们的研究工作。

This grant provides funding to study efficient algorithms for solving stochastic and large-scale convex programming (CP) problems. The development of CP algorithms is now facing a few new challenges. First, the incorporation of uncertainty, and possibly dynamics, has considerably increased the difficulty of solving CP. Second, the size of CP problems arising from many emerging applications, such as statistical learning, far exceeds the capability of existing polynomial-time solvers. In this project, we will study a new and promising class of stochastic first-order methods for solving stochastic CP and establish their convergence and large-deviation properties. Moreover, using these stochastic methods as the driving-force, we aim at enhance our capability for solving extremely large-scale CP through the development of novel randomization schemes. Finally we will advance the optimization solvers for various statistical learning models which are crucial in order to transform data into knowledge in our information age. If successful, the results of this research will lead to algorithmic innovation for solving Large-scale and stochastic CP problems arising from diverse areas in logistics, finance, energy, medicine, defense and science. In green energy systems, stochastic programming models are used for long-term operations planning of hydro-thermal power systems to deal with the stochastic streamflows. In medicine, statistical learning techniques can be applied to diagnose breast cancer by utilizing characteristics of individual cells to discriminate benign from malignant breast lumps. These are just a few application examples of SP, large-scale CP and statistical learning problems to be studied in this research that will impact society. Additionally, this research will contribute to open research infrastructure in optimization through the development of CP-solver software. We will also engage students in our research efforts.