. We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, first-level constraints are linear, and second-level (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, ill-conditioning, convexity of the objective, monotonicity and symmetry of the second-level problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.maths.mu.OZ.AU/~danny/qpecgendoc.h...

In the paper, the question is investigated if a bundle algorithm can be used to compute approximate solutions for bilevel programming problems where the lower level optimal solution is in general not uniquely determined. To give a positive answer to this question, an appropriate regularization approach is used in the lower level. In the general case, the resulting algorithm computes an approximate solution. If the problem proves to have strongly stable lower level solutions for all parameter values in a certain neighborhood of the stationary solutions of the bilevel problem, convergence to stationary solutions can be shown. Keywords: bilevel programming, parametric optimization, bundle algorithm, nondifferentiable optimization 1

Let x 0 be a locally optimal solution of a smooth parametric nonlinear optimization problem minff(x; y) : g(x; y) 0; h(x; y) = 0g for a fixed value y = y 0 . If the strong sufficient optimality condition of second order together with the Mangasarian-Fromowitz and the constant rank constraint qualifications are satisfied, then x 0 is strongly stable in the sense of Kojima and the corresponding function of locally optimal solutions of perturbed problems is locally Lipschitz continuous. In the paper we give one tool for computing its generalized Jacobian. We also show that this function is quasidifferentiable in the sense of Dem'yanov and Rubinov and describe one quasidifferential. In the last part of the paper a method is developed for reducing the quasidifferential which is especially useful when both the super- and the subdifferential are spanned by finitely many elements as in our case. MR 1991 Subject Classification: 90C31, 49K40 1

If a strong sufficient optimality condition of second order together with the Mangasarian-Fromowitz and the constant rank constraint qualifications are satisfied for a parametric optimization problem, then a local optimal solution is strongly stable in the sense of Kojima and the corresponding optimal solution function is locally Lipschitz continuous. In the article the possibilities for the computation of subgradients of this function are discussed. We will give formulae for the guaranteed computation of the entire subdifferential, provided that an additional assumption is satisfied. An example will show the necessity of this assumption. Moreover, this assumption is difficult to be verified. Without it, a subgradient can be computed with non-polynomial complexity in the worst case. A last approach yields a subgradient with probability one in polynomial time. 1

The bilevel programming problem (BLPP) is equivalent to a two-person Stackelberg game in which the leader and follower pursue individual objectives. Play is sequential and the choices of one affect the choices and attainable payoffs of the other. The purpose of this paper is to investigate an extension of the linear BLPP where both players' objective functions are bilinear. To overcome certain discontinuities in the master problem, a regularized term is added to the follower's objective function. Using ideas from parametric programming, the directional derivatives of the regularized follower's solution function are computed along with its generalized Jacobian. This allows us to develop a bundle trust region algorithm. Theoretical results related to the existence of solutions are presented as well as a convergence analysis of the proposed methodology. Key words: bilevel programming, bundle algorithm, Lipschitz continuity, generalized gradients, nondifferentiable optimization. 1 Freiberg University of Mining and Technology, Germany, dempe@math.tu-freiberg.de 2 Graduate Program in Operations Research, University of Texas, Austin, U.S.A., jbard@mail.utexas.edu 1