Hierarchical Optimization Under Uncertainty

Stochastic bilevel optimization is a new mathematical paradigm for optimal decision making problems under uncertainty where some decision makers are in a hierarchical relationship.
The hierarchical structure appears in many real-world situations, where there is an interaction between two or more interdependent decision makers/systems some of which have certain “privilege” for the so-called leader, i.e., a strategic market player, who decides first, and the other decision makers in the chain known as the followers, have to subsequently react in a manner that suits their individual interests.

The PhD project will focus on two-stage stochastic bilevel optimization problems where upper level decision is taken before realization of uncertainty whereas lower level decisions are made after observation of uncertainty and upper level decision. Extension to multistage is possible. Depending on availability of information on on uncertainty, we may consider a stochastic or distributionally robust model.

An important approach to tackle this kind of mathematical problem is mathematical programs with equilibrium constraints and value function-type reformulations of the problem will then be considered and their theoretical properties, including the relationship between the approximated and the true problem, existence of optimal results and optimality conditions will be studied; computational methods will need to be developed by using tools from nonsmooth and nonconvex optimization. Special attention will be given to numerical methods based be on the well known value function approach, for which techniques from variational analysis will be exploited to construct accurate approximations. Applications in energy and transportation are envisaged.