Bilevel programming (see Bilevel programming: formulation) is ideally suited to model situations where the decision maker does not have full control over all decision variables. Five such situations are described in this article. The rst example involves the improvement of a road network through either capacity expansion , traac signals synchronization, vehicle guidance systems, etc. While mana...

Abstract: In this paper, we study a pessimistic linear bilevel multi-follower programming problem in a referential-uncooperative situation (for short, PLBMF-RU) where the followers are uncooperative while having cross reference to decision making information of the other followers. We first give a model and the solution definitions of PLBMF-RU, and then use a simple example to illustrate it. Fu...

We consider a bilevel programming problem. The leader’s objective function is assumed to be linear. The follower’s problem is a quantile minimization problem. It is assumed that the follower’s loss function is bilinear. We obtain a deterministic equivalent of the original problem in the case of a scalar random variable. In the case of the normal distribution of the random vector, an algorithm t...

We consider equilibrium constrained optimization problems, which have a general formulation that encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated KKM lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topol...

An alternative definition of the linear bilevel programming problem BLP has recently been proposed by Lu, Shi, and Zhang. This note shows that the proposed definition is a restriction of BLP . Indeed, the new definition is equivalent to transferring the first-level constraints involving second-level variables into the second level, resulting in a special case of BLP in which there are no first-...

In this paper we propose a mixed integer bilevel problem having a probabilistic knapsack constraint in the first level. The problem formulation is mainly motivated by practical pricing and service provision problems as it can be interpreted as a model for the interaction between a service provider and clients. We assume the probability space to be discrete which allows us to reformulate the pro...

In this paper, we consider the problem of network design for hazardous material transportation where the government designates a network, and the carriers choose the routes on the network. We model the problem as a bilevel network flow formulation and analyze the bilevel design problem by comparing it to three other decision scenarios. The bilevel model is difficult to solve and may be ill-pose...

Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the constraints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In th...