Stephan Dempe

Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints

In this bibliography main directions of research as well as main fields of applications of bilevel programming problems and mathematical programs with equilibrium constraints are summarized. Focus is also on the difficulties arising from nonuniqueness of lower-level optimal solutions and on optimality conditions.

Is bilevel programming a special case of a mathematical program with complementarity constraints

Bilevel programming problems are often reformulated using the Karush–Kuhn–Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is “No” even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater’s constraint qualification. We also show by examples that this correspondence can fail if the Slater’s constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.

Directional derivatives of the solution of a parametric nonlinear program

Consider a parametric nonlinear optimization problem subject to equality and inequality constraints. Conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known. We show, under the additional assumption of constant rank of the active constraint gradients, that the optimal solution is actually piecewise smooth, hence B-differentiable. We show, for the first time to our knowledge, a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique.

A necessary and a sufficient optimality condition for bilevel programming problems

A necessary and a sufficient condition for local optimal solutions of bilevel programming problems are developed using differential stability results for parametric optimization problems. Verification of these conditions reduces to the solution of some auxiliary combinatorial optimization problems.

New necessary optimality conditions in optimistic bilevel programming

The article is devoted to the study of the so-called optimistic version of bilevel programming in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation. Considering a basic optimistic model in bilevel programming, we reduce it to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model. Using advanced formulas for computing basic subgradients of value/marginal functions in variational analysis, we derive new necessary optimality conditions for bilevel programs reflecting significant phenomena that have never been observed earlier. In particular, our optimality conditions for bilevel programs do not depend on the partial derivatives with respect to parameters of the smooth objective function in the parametric lower-level problem. We present efficient implementations of our approach and results obtained for bilevel programs with differentiable, convex, linear, and Lipschitzian functions describing the initial data of the lower-level and upper-level problems. ¶This work is dedicated to the memory of Prof. Dr Alexander Moiseevich Rubinov.

Optimality conditions for bilevel programming problems

Focus in the paper is on optimality conditions for bilevel programming problems. We start with a general condition using tangent cones of the feasible set of the bilevel programming problem to derive such conditions for the optimistic bilevel problem. More precise conditions are obtained if the tangent cone possesses an explicit description as it is possible in the case of linear lower level problems. If the optimal solution of the lower level problem is a PC 1-function, sufficient conditions for a global optimal solution of the optimistic bilevel problem can be formulated. In the second part of the paper relations of the bilevel programming problem to set-valued optimization problems and to mathematical programs with equilibrium constraints are given which can also be used to formulate optimality conditions for the original problem. Finally, a variational inequality approach is described which works well when the involved functions are monotone. It consists in a variational re-formulation of the optimality conditions and looking for a solution of the thus obtained variational inequality among the points satisfying the initial constraints. A penalty function technique is applied to get a sequence of approximate solutions converging to a solution of the original problem with monotone operators.

A simple algorithm for the-linear bilevel programming problem

In the Paper an algorithm for the linear bilevel programming problem is offered which applies mainly the ususal simplex method with and additional rule for including slack variables into the basis. The algorithm bases on the full description of the feasible set in a neighbourhood of a feasible point. This description is obtained using the theory of subradients as well as the concept of “active constraints”. The result is an algorithm which seems to be easier to implement as other published procedures also based on theorem that every solvable linear bilevel programming problem has a basic solution.

Discrete bilevel programming: Application to a natural gas cash-out problem

In this paper, we present a mathematical framework for the problem of minimizing the cash-out penalties of a natural gas shipper. The problem is modeled as a mixed-integer bilevel programming problem having one Boolean variable in the lower level problem. Such problems are difficult to solve. To obtain a more tractable problem we move the Boolean variable from the lower to the upper level problem. The implications of this change of the problem are investigated thoroughly. The resulting lower level problem is a generalized transportation problem. The formulation of conditions guaranteeing the existence of an optimal solution for this problem is also in the scope of this paper. The corresponding results are then used to find a bound on the optimal function value of our initial problem.

Necessary optimality conditions in pessimistic bilevel programming

This article is devoted to the so-called pessimistic version of bilevel programming programs. Minimization problems of this type are challenging to handle partly because the corresponding value functions are often merely upper (while not lower) semicontinuous. Employing advanced tools of variational analysis and generalized differentiation, we provide rather general frameworks ensuring the Lipschitz continuity of the corresponding value functions. Several types of lower subdifferential necessary optimality conditions are then derived by using the lower-level value function approach and the Karush–Kuhn–Tucker representation of lower-level optimal solution maps. We also derive upper subdifferential necessary optimality conditions of a new type, which can be essentially stronger than the lower ones in some particular settings. Finally, certain links are established between the obtained necessary optimality conditions for the pessimistic and optimistic versions in bilevel programming.

The bilevel programming problem: reformulations, constraint qualifications and optimality conditions

We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.