Structured Stochastic Nonconvex Optimization

在全球范围内，可再生能源已成为减少温室气体排放的重要途径。加拿大是可再生能源领域的全球领先者，根据CanWEA的数据，加拿大拥有世界第九大风力发电机队。虽然对可再生能源的投资在增加，但在有效利用它们方面仍然存在挑战。这类挑战的一个重要方面是存在每天几次更新的风能滚动预报。安大略省风力发电场的能源预测存在重大误差，尽管他们的48小时预测每小时更新一次。同样的问题影响到许多库存问题，包括用于公共卫生的疫苗库存。当前使用确定性预测的标准滚动水平程序不能适应不确定性，并且使用随机预测非常耗时。这项研究计划将探索一种新的方法来应对前述滚动预报的挑战。我们的方法利用了确定性前瞻模型的结构简单性，同时通过在标准滚动水平过程中添加可调参数来允许更多的灵活性。它可以很容易地处理更新预报，并且对预报的质量不敏感，但需要通过求解一个非凸优化问题来优化策略。近年来，非凸模型也因其能更好地模拟现实和拟合数据点而被广泛应用于数据分析和机器学习问题。深度神经网络是许多函数的组合，因此通常是非凸的，因此在计算机视觉、自动驾驶汽车和医疗保健等复杂系统的建模方面吸引了相当大的兴趣。然而，训练这样的模型仍然是一个研究挑战。非凸问题也出现在其他几个应用中，例如库存管理问题、量子优化和机器学习中的超参数调整，在这些应用中，只有来自目标函数的观测可用。所有上述问题都主要是非凸的，这意味着它们比在有限的现实世界应用中出现的表现良好的凸函数更难求解。这一研究计划将导致开发具有理论收敛保证的高效算法来解决上述类别的问题。这些算法的实际性能也将通过真实的基准数据集进行评估。然后，这些算法可以用于在存在不确定性的情况下解决广泛的现实世界问题。它们可以更好地利用可再生能源，从而为加拿大更多的家庭提供电力。这种发展需要本科生和研究生两级的受训人员，他们将接受培训，以获得用于建模和解决不确定情况下的复杂决策问题的分析工具。

Globally, renewable energy has become an important way to reduce greenhouse gas emissions. Canada is a global leader in renewable energy which is home to the world's ninth largest wind-generating fleet according to CanWEA. While the investments in renewable energy are growing, there are still challenges in efficiently using them. An important aspect of such challenges is the presence of the rolling forecasts of wind energy updated as frequently as several times per day. There are significant errors in energy forecasts from Ontario wind farms, despite the fact that their 48-hour forecasts are updated hourly. The same issue affects many inventory problems including vaccine inventories for public health. The current standard rolling horizon procedures using deterministic forecasts do not accommodate uncertainty and using stochastic lookahead is very time-consuming. This research program will investigate a new approach for the aforementioned challenges of rolling forecasts. Our approach draws on the structural simplicity of deterministic lookahead models while allows more flexibility by adding tunable parameters to the standard rolling horizon procedures. It can then easily handle the updating forecasts and is not sensitive to their quality, however, requires optimizing policies through solving a non-convex optimization problem. Recently, non-convex models have also been widely used in data analysis and machine learning problems due to their ability in better modelling of the reality and fitting to the data points. Deep neural networks, which are the composition of many functions and thus are non-convex in general, have attracted considerable interest in modelling complex systems such as computer vision, self-driving cars, and healthcare. However, training such models is still a research challenge. Non-convex problems also arise in several other applications such as inventory management problems, quantum optimization, and hyper parameter tuning in machine learning in which only observations from an objective function are available. All of the above-mentioned classes of problems are mainly non-convex implying that they are much harder to solve than the well-behaved convex functions arising in limited real-world applications. This research program will lead to the development of efficient algorithms with theoretical convergence guarantees for solving the aforementioned classes of problems. The practical performance of these algorithms will also be assessed by real benchmarking data sets. These algorithms can be then used to solve a wide range of real-world problems under the presence of uncertainty. They can be beneficial in better utilization of renewable energy and so delivering electricity power to more homes across Canada. Such developments require trainees both in undergraduate and graduate levels, who will be trained to gain analytical tools used for modelling and solving complex decision-making problems under uncertainty.