Bilevel optimization problems arise in many ML and signal processing applications, where the aim is to find the minimal norm or most sparse optimal solution of an underdetermined optimization problem. Traditionally, these problems have been solved by regularization which requires tuning of the regularization parameter. We are focus on an alternative approach which utilizes first order optimization methods to directly solve this problem, for which we provide rate of convergence guarantees.
Adaptive planning radiotherapy treatment based on inaccurate and evolving bio-marker information collected from imaging during the treatment. Radiotherapy plan is composed on the amount and angle of radiation in each stage of the treatment, where the goal is to get the maximal dose to the tumor while protecting healthy organs. The challenge comes from the resulting problem being a large-scale mixed integer problem, and the dependence between optimal decision and the future bio-marker levels.
Multi-stage linear stochastic optimization problems are known to be challenging. An added difficulty arises when the distribution of the uncertainty is not known exactly, and alternatively only historical sample paths of the problem are available. We explore solving this problem by using data-driven distributionally robust optimization, for which we provide convergence guarantees. Additionally, we explore solving the resulting optimization problem by approximation methods.