Solving ill-posed bilevel programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem

Comparing reinforcement learning approaches for solving game theoretic models: a dynamic airline pricing game example

Due to the difficulty in solving game theoretic models, there is a tendency to focus on the overly simplistic dynamic airline pricing games or to even ignore competition completely. Recent changes to the industry mean that airlines can no longer ignore competitors in their model. This paper adds more complex customer model aspects; i.e., customer choice using a Logit model, customer demand using a linear probabilistic demand model, and market size using a binary random function; into an existing solvable game that only had a simple customer model. A reinforcement learning method was used to solve the newly formed games with mixed results.<br/