Section: New Results

Bilevel Programming

Bilevel approaches for energy management problems: We have proposed the first bilevel pricing models to explore the relationship between energy suppliers and customers who are connected to a smart grid. Due to their definition, bilevel models enable to integrate customer response into the optimization process of supplier who aims to maximize revenue or minimize capacity requirements. In our setting, the energy provider acts as a leader (upper level) that takes into account a smart grid (lower level) that minimizes the sum of users' disutilities. The latter bases its decisions on the hourly prices set by the leader, as well as the schedule preferences set by the users for each task. The pricing problems, we model, belong to the category of single leader single follower problems. Considering both the monopolistic and competitive environment we present two bilevel bilinear bilinear problems with continuous variables. Heuristics solutions methods are defined to solve large size instances of the models. They are based on the interactions between prices, schedules and peaks. Numerical results on randomly generated instances illustrate numerically the validity of the approach, which achieves an `optimal trade-off between three objectives: revenue, user cost, and peak demand. Moreover, they put into highlight the ability of the heuristics to produce high quality results compared to the solution of MIP reformulations of the models[36].

New formulations for solving Stackelberg games: We analyzed general Stackelberg games (SGs) and Stackelberg security games (SSGs). SGs are hierarchical adversarial games where players select actions or strategies to optimize their payoffs in a sequential manner. SSGs are a type of SGs that arise in security applications, where the strategies of the player that acts first consist in protecting subsets of targets and the strategies of the followers consist in attacking one of the targets. We review existing mixed integer optimization formulations in both the general and the security setting and present new formulations for the the second one. We compare the SG formulations and the SSG formulations both from a theoretical and a computational point of view. We indentify which formulations provide tighter linear relaxations and show that the strongest formulation for the security version is ideal in the case of one single attacker. Our computational experiments show that the new formulations can be solved in shorter times [46].