Combined Reformulation of Bilevel Programming Problems

More documents

Recommendations

Info

72Of course it is difficult to check if one point is local optimal for a combinedreformulationforeveryλ ∈ Λ(¯x,ȳ). Itisevenpossibletofindanexample(cf. [6]withconvexlowerlevel problem)such that foronly oneKKT multiplier ˆλ ∈ intΛ(¯x,ȳ)thepoint (¯x,ȳ,ˆλ) is not local optimal for the combined reformulation. Hence it is alsonot a local solution of the initial bilevel problem. For that reason it is importantto consider, wether it is possible to restrict the set of regarded KKT multipliersunder appropriate assumptions. Some regularity conditions such as Constant RankConstraint Qualification (CRCQ) (cf. [9]) are helpful in order to achieve a betterresult. We can state the following conclusion similar to the result from [6].Corollary 1. Assume that at the point (¯x,ȳ), ȳ ∈ Ψ(¯x) both MFCQ and CRCQare satisfied for the lower level problem. Moreover, let (¯x,ȳ,¯λ) be a local solutionof the combined reformulation (5) for all vertices ¯λ of the set of KKT multipliersΛ(¯x,ȳ). Then the point (¯x,ȳ) is a local solution of the bilevel programming problem.If we consider Example 2, then we can state, that both MFCQ an CRCQ aresatisfied at the point (0,1) for the lower level problem and that is why we need onlyto check if two points (0,1,(0,3))and (0,1,(3,0))are local solutions of the combinedreformulation. We can easily state, that the point (0,1,(3,0)) is not local optimalfor the new reformulation and consequently it is also not local solution of the initialbilevel programming problem.3. Combined reformulation as a nonsmooth MPECAfter equivalence considerations we now need to find out which constraint qualificationscan be satisfied for this problem and how to define the necessary optimalityconditions.Sincecombined reformulationhasastructureofanonsmoothMPEC(see Section1) it is possible to apply results from Movahedian and Nobakhtian [13] to obtainsome regularity and stationarity conditions.At first we define the combined reformulation in disjunctive form (cf. [13,8]):F(x,y) → minx,y,λH(x,y,λ) ∈ Γ,(12)where all restrictions are aggregated to one function:H(x,y,λ) :=(G IG (x,y),f(x,y)−V(x),∇ y f(x,y)+p∑λ i ∇ y g i (x,y),i=1−g α (x,y),λ α ,−g β (x,y),λ β ,−g γ (x,y),λ γ), (13)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
73whereI G := I G (¯x,ȳ) := {i = 1,...,k : G i (¯x,ȳ) = 0}and the right hand side of the constraints is defined as follows:⋃Γ := Γ β1,β 2. (14)(β 1,β 2)∈P(β)In order to define the complementarity constraint λ ⊤ g(x,y) = 0 we need tointroduce the following index sets:α := α(¯x,ȳ,¯λ) := { i : g i (¯x,ȳ) = 0, ¯λ i > 0 } ,β := β(¯x,ȳ,¯λ) := { i : g i (¯x,ȳ) = 0, ¯λ i = 0 } , (15)γ := γ(¯x,ȳ,¯λ) := { i : −g i (¯x,ȳ) > 0, ¯λ i = 0 } .The index set β, for which both g i (x,y) as well as λ i are active, can be dividedinto two partitions in many various ways. The set of all partitions of β is defined asfollows:P(β) := {(β 1 ,β 2 ) : β 1 ∪β 2 = β, β 1 ∩β 2 = ∅}. (16)With the aid of this set we can define the convex polyhedra:Γ β1,β 2:= R |IG|−×R − ×0 m ×0 |α| ×R |α|+ ×∆ β1,β 2×∆ β2,β 1×R |γ|+ ×0 |γ| , (17)where |A| denotes the cardinality of the set A and{0 : j ∈ µ,(∆ µ,ν ) j :=R + : j ∈ ν.Notice that this disjunctive problem depends on vector (¯x,ȳ,¯λ) and hence it isequivalentto the combined reformulation(i.e. the feasible sets ofthese twoproblemsare equal) in a neighbourhood of the considered point (¯x,ȳ,¯λ).ThereareseveralconstraintqualificationslikeLICQorMFCQ,thatwereadaptedfor MPEC (cf. [7]): MPEC-LICQ, MPEC-SMFCQ, piecewise MPEC-MFCQ. Thelatter one is also the weakest of this group of regularity conditions. For this reasonwe define this condition for a nonsmooth MPEC (see [10]):Definition 1. Consider the following optimization problem:(18)f(z) −→ minz(19)G i (z) = 0, i ∈ α, H i (z) = 0 i ∈ γ,G i (z) ≥ 0, H i (z) = 0 i ∈ β 1 ,G i (z) = 0, H i (z) ≥ 0 i ∈ β 2 ,g(z) ≤ 0, h(z) = 0.where f,g,G,H are local Lipschitz continuous. The piecewise MPEC-NMFCQ holdsat the feasible point ¯z if NMFCQ is satisfied at this point for every partition (β 1 ,β 2 ) ∈P(β).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
Page 2 and 3: 66problem, which is in general a se
Page 7: 71Example 2. Consider following (no
Page 11 and 12: 75• Linearized tangent cone:T lin
Page 13 and 14: 772. MPEC strong-stationary, if the
Page 15 and 16: 79[2] Bard J.F.; Practical Bilevel

Combined Reformulation of Bilevel Programming Problems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?