This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).

This paper provides an introductory survey of a class of optimization problems known as bilevel programming. We motivate this class through a simple application, and then proceed with the general formulation of bilevel programs. We consider various cases (linear, linear-quadratic, nonlinear), describe their main properties and give an overview of solution approaches.

Abstract. We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upper-level objective and all constraintdefining functions, as well as a quadratic approximation of the lower-level objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.

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We present an algorithm for solving nonlinear programming problems involving a partially separable objective function whose derivatives are assumed to be unavailable. At each iteration, we construct a quadratic interpolation model of the objective function around the current iterate and minimize this model to obtain a trial step. The whole process is embedded within a trust-region framework. We further propose to use ideas of Curtis, Powell and Reid to minimize the number of calls to the objective function in the part of the derivative-free algorithm that improves the geometry of the interpolation set. Numerical experiments tend to confirm the promising behaviour of the algorithm.