Advanced Topics in Optimization

This course provides a rigorous introduction to dynamic optimization. The course is structured in three parts. Part I covers the fundamentals of dynamic programming (DP), focusing on Bellman's principle of optimality and its application to discrete-time finite and infinite horizon problems, including Markov Decision Processes (MDPs). Part II transitions to online optimization, where optimal decisions must be made sequentially with incomplete future information. Topics include competitive analysis, online primal-dual methods, and applications. Part III discusses recent research articles that use these frameworks in a variety of applications of interest, mostly drawing on examples from operations research, computer science, and economics.