A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

An overview of Stackelberg pricing in networks

The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bi-linear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This manuscript contains a compilation of theoretical and algorithmic results on the Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then, we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and the development of algorithms: establishing NP-hardness, approximability, and polynomially solvable cases, and an efficient exact branch-and-bound algorithm.mathematical applications;

An overview of Stackelberg pricing in networks

The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bilinear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This paper contains a compilation of theoretical and algorithmic results on the network Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and for the development of algorithms: establishing NP-hardness, approximability, special cases solvable in polynomial time, and an efficient exact branch-and-bound algorithm.Economics ;

Data-driven Inverse Optimization with Imperfect Information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches

Three essays on bilevel optimization algorithms and applications

This thesis consists of three journal papers I have worked on during the past three years of my PhD studies. In the first paper, we presented a multi-objective integer programming model for the gene stacking problem. Although the gene stacking problem is proved to be NP-hard, we have been able to obtain Pareto frontiers for smaller sized instances within one minute using the state-of-the-art commercial computer solvers in our computational experiments. In the second paper, we presented an exact algorithm for the bilevel mixed integer linear programming (BMILP) problem under three simplifying assumptions. Compared to these existing ones, our new algorithm relies on weaker assumptions, explicitly considers infinite optimal, infeasible, and unbounded cases, and is proved to terminate infinitely with the correct output. We report results of our computational experiments on a small library of BMILP test instances, which we have created and made publicly available online. In the third paper, we presented the watermelon algorithm for the bilevel integer linear programming (BILP) problem. To our best knowledge, it is the first exact algorithm which promises to solve all possible BILPs, including finitely optimal, infeasible, and unbounded cases. What is more, our algorithm does not rely on any simplifying condition, allowing even the case of unboundedness for the high point problem. We prove that the watermelon algorithm must finitely terminate with the correct output. Computational experiments are also reported showing the efficiency of our algorithm