Elsevier

Operations Research Letters

Volume 29, Issue 4, November 2001, Pages 171-179
Operations Research Letters

A trust region algorithm for nonlinear bilevel programming

https://doi.org/10.1016/S0167-6377(01)00092-XGet rights and content

Abstract

We propose to solve generalized bilevel programs by a trust region approach where the “model” takes the form of a bilevel program involving a linear program at the upper level and a linear variational inequality at the lower level. By coupling the concepts of trust region and linesearch in a novel way, we obtain an implementable algorithm that converges to a strong stationary point of the original bilevel program.

Section snippets

A trust region approach to bilevel programming

In this paper we consider bilevel programs (or MPECs, i.e., mathematical programs with equilibrium constraints, see [5]) of the formminx∈X,y∈Y(x)f(x,y)s.t.〈F(x,y),y−y′〉⩽0∀y′∈Y(x),where the mapping F is strongly monotone with respect to the lower level variable y and where the sets X and Y(x)={y:Ax+By⩾b} are polyhedral. We propose for its solution a trust region approach where the model is itself a bilevel program of a combinatorial nature that can be solved for a global optimum.

This approach,

Local behavior of y(x) and ȳ(x)

Theorem 1

The functions y(x) and ȳ(x) are Lipschitz continuous on their common domain of definition D={x∈X:Y(x)≠∅}.

Proof

Under local Lipschitz and strong monotonicity assumptions on the mapping F, Dafermos [2] has shown that the reaction function y(x) is Lipschitz continuous, provided that the projection operator pz(x)=projY(x)(z) is also Lipschitz continuous, for fixed z. By definition, we have thatpz(x)=argminy∈Y(x)12||y−z||2or equivalently, pz(x) satisfies the generalized equation0∈pz(x)−z+∂δ(Y(x)),where δ

Approximation results

In this section we prove that, applied to an iterate sufficiently close to a point which is not a strong stationary point, i.e., a point where at least one feasible descent direction exists, algorithm BlTrust will generate a new iterate with an objective value strictly less than that of the current iterate. The proof of this result is based on three approximation lemmas, which make use of the notation described in Table 1.

Lemma 1

||y(xε)−ȳ(xε)||⩽σε2 and ||y(x̄ε)−ȳε||⩽σε2.

Proof

By definition of y(xε) and ȳ

Convergence analysis

We say that a feasible point x is strongly stationary for a function g if there do not exist a positive number α and a sequence {xi}iI converging to x such thatg(xi)<g(x)−α||x−xi||∀i∈I.Note that the set of strong stationary points may be a strict subset of points that are stationary in the Clarke sense, i.e., points for which the zero vector belongs to the generalized subdifferential. Whenever the function g is directionally differentiable over its domain, the set of strong stationary points

Conclusion

Although algorithm BlTrust is only guaranteed to generate B-stationary limit points, it has the capability to move away from nonglobal stationary points. In this respect, it is unlikely to get trapped into “weak” local solutions and holds the potential of solving to global optimality generalized bilevel programs involving mildly nonlinear functions f and F. For that reason we refer to BlTrust as a “semi-global” method.

Future work on BlTrust will focus on its implementation and on relaxing some

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