## 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.
Moreover the follower is able to produce renewable energy and store it. The pricing problems, we model, belong to the category of stochastic single leader single follower problems. A scenario based approach is based to solve the problem. For each scenario, the bilevel program is solved by rewriting it as a single level optimization program.
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 [53].

**Network pricing problems with unit toll:**
In the so-called network pricing problem an authority owns some arcs of the network and tolls them, while users travel between their origin and destination choosing their minimum cost path. We consider a unit toll scheme, and in particular the cases where the authority is imposing either the same toll on all of its arcs, or a toll proportional to a given parameter particular to each arc (for instance a per kilometer toll). We
show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial, proposing ad-hoc solution algorithms and relating these problems to the parametric shortest path problem. We then address a robust approach using an interval representation to take into consideration uncertainty on parameters. We show how to modify the algorithms for the deterministic case to solve the robust counterparts, maintaining their complexity class [15].

**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 identify 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 [66].

**A branch and price algorithm for solving Stackelberg Security games:**
Mixed integer optimization formulations are an attractive alternative to solve Stackelberg Game problems
thanks to the efficiency of state of the art mixed integer algorithms. In particular, decomposition
algorithms, such as branch and price methods, make it possible to tackle instances large enough to represent
games inspired in real world domains. We focus on Stackelberg Games that arise from a security application and investigate the
use of a new branch and price method to solve its mixed integer optimization formulation. We prove that
the algorithm provides upper and lower bounds on the optimal solution at every iteration and investigate
the use of stabilization heuristics. Our preliminary computational results compare this solution approach
with previous decomposition methods obtained from alternative integer programming formulations of
Stackelberg games [29].

**A new general-purpose algorithm for mixed-integer bilevel linear programs:**
We considered bilevel problems with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We proposed a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study was presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperformed (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solved to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver was made publicly available online
[18], [19].

**A mixed-integer programming based heuristic for generalized interdiction problems:**
We considered a subfamily of mixed-integer linear bilevel problems that we call Generalized Interdiction Problems. This class of problems includes, among others, the widely-studied interdiction problems, i.e., zero-sum Stackelberg games where two players (called the leader and the follower) share a set of items, and the leader can interdict the usage of certain items by the follower. Problems of this type can be modeled as Mixed-Integer Nonlinear Programming problems, whose exact solution can be very hard. We propose a new heuristic scheme based on a single-level and compact mixed-integer linear programming reformulation of the problem obtained by relaxing the integrality of the follower variables. A distinguished feature of our method is that general-purpose mixed-integer cutting planes for the follower problem are exploited, on the fly, to dynamically improve the reformulation. The resulting heuristic algorithm proved very effective on a large number of test instances, often providing an (almost) optimal solution within very short computing times.
[20]

**Unit Commitment under Market Equilibrium Constraints:** Traditional
(deterministic) models for the Unit Commitment problem (UC) assume that the net
demand for each period is perfectly known in advance, or in more recent and
more realistic approaches, that a set of possible demand scenarios is known
(leading to stochastic or robust optimization problems). However, in practice,
the demand is dictated by the amounts that can be sold by the producer at given
prices on the day-ahead market. We modeled and solved the UC problem with a
second level of decisions ensuring that the produced quantities are cleared at
market equilibrium. In its simplest form, we are faced to a bilevel
optimization problem where the first level is a MIP and the second level
linear. As a first approach to the problem, we assumed that demand curves
and offers of competitors in the market are known to the operator. Following the classical approach for these models, we turned the problem into a single-level program by rewriting and
linearizing the first-order optimality conditions of the second level [50].