Structured Stochastic Nonconvex Optimization

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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.