Combined Reformulation of Bilevel Programming Problems

78Because of the fact, that the optimal solution is a separated point with respectto all variables, the tangent cone consists only of a zero vector. Now we need todetermine the linearizedcone by solvingasystem of linearinequalities and equalities(23) with α = γ = ∅ and β = {1,2}:T linMPEC (¯x,ȳ,¯λ) = { (d 1 ,d 2 ,v 1 ,v 2 ) ∈ R 4 : −d 2 ≤ 0,v 1 +v 2 = 0,d 1 −d 2 ≥ 0, v 1 ≥ 0, v 1 (d 1 −d 2 ) = 0−d 1 −d 2 ≥ 0, v 2 ≥ 0, v 2 (−d 1 −d 2 ) = 0} ={(d1 ,d 2 ,v 1 ,v 2 ) ∈ R 4 : d 1 = d 2 = v 1 = v 2 = 0 } .That means: T B ((¯x,ȳ,¯λ), H −1 (Γ)) = T linMPEC (¯x,ȳ,¯λ).Because of the fact, that in this example the optimal value function is differentiable,we can state, that the condition (26) is satisfied (see [13] for more details).Consequently MPEC-NACQ holds at the point (¯x,ȳ,¯λ).If we consider MPEC-NGCQ we find out, that because of the definition of theFréchet normal cone (25) and due to the fact, that T B ((¯x,ȳ,¯λ), H −1 (Γ)) = {0}, thecondition (27) is trivially satisfied and the condition MPEC-NGCQ holds also at theconsidered point. Consequently MPEC-WACQ is also satisfied at (¯x,ȳ,¯λ).Therefore we can state, that the point (¯x,ȳ,¯λ) is MPEC M-stationary with e.g.the following KKT multipliers (λ G ,λ V ,λ KKT ,λ g ,λ λ ) = (0,0,0,1,1,0,0).4. ConclusionThe combined reformulation is a convenient proposal how to deal with bilevel optimizationproblems if the lower level problem is not assumed to be convex. It ispossible to find some regularity conditions and to derive optimality conditions, thatcan be satisfied for this nonsmooth MPEC. The obtained stationarity conditionswithout using partial calmness are actually the same as with using this regularitycondition (cf. [19]). In the future it would be interesting to examine whether thereexists a relationship between partial calmness and MPEC-NACQ, MPEC-WACQ orMPEC-NGCQ.5. References[1] Bank B., Guddat J., Klatte D., Kummer B., Tammer K.; Non-Linear ParametricOptimization, Akademie-Verlag, Berlin 1982.Publikacja objęta jest prawem autorskim. Wszelkie prawa zastrzeżone. Kopiowanie i rozpowszechnianie zabronione.Publikacja przeznaczona jedynie dla klientów indywidualnych. Zakaz rozpowszechniania i udostępniania w serwisach bibliotecznych