Optimality conditions for pessimistic bilevel problems using convexificator
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Positivity
Optimality conditions for pessimistic bilevel problems using convexificator S. Dempe1 · N. Gadhi2 · L. Lafhim2 Received: 1 August 2018 / Accepted: 2 February 2020 © Springer Nature Switzerland AG 2020
Abstract The purpose of this paper is to study the pessimistic version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data). By using optimal value function, we transform the initial problem into a generalized minimax optimization problem. Using convexificators, first-order necessary optimality conditions are then established. An example that illustrates our findings is also given. Keywords Optimization and variational analysis · Pessimistic bilevel programs · Two-level value functions · Sensitivity analysis · Generalized differentiation · Optimality conditions Mathematics Subject Classification 90C26 · 90C30 · 90C31 · 90C46
1 Introduction The paper is devoted to the study of necessary optimality conditions for a pessimistic bilevel programming in finite-dimensional spaces. A general problem of this type, called two level programming, can be written as (P) :
B
“min” {F(x, y) : G i (x) ≤ 0, ∀i ∈ I = {1, . . . , p}, y ∈ S(x)} x
(1)
N. Gadhi [email protected] S. Dempe [email protected] L. Lafhim [email protected]
1
Department of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Freiberg, Germany
2
LAMA, Department of Mathematics, Sidi Mohamed Ben Abdellah University, Dhar El Mahraz, Fez, Morocco
123
S. Dempe et al.
where, for each x ∈ Rn , S(x) is the solution set of the following parametric optimization problem (Px ) :
min{ f (x, y) : g j (x, y) ≤ 0, ∀ j ∈ J = {1, . . . , q}} y
(2)
where, n ≥ 1 and m ≥ 1 are integers, F, f , G i , g j : Rn × Rm → R are functions for all i ∈ I and j ∈ J . For simplicity, we denote the upper and the lower-level constraint sets respectively by X = {x ∈ Rn |G i (x) ≤ 0, ∀i ∈ I } and (x) = {y ∈ Rm |g j (x, y) ≤ 0, ∀ j ∈ J }. Generally speaking, the framework of bilevel optimization problems is that there are both leader-level and follower-level problems, where the upper level (leader) intends to minimizing his/her cost function F with respect to the variable x while taking into account the reaction y of the lower level (follower). In the upper level minimization problem (P), quotation marks have been used because of an ambiguity that arises when the solution set S(x) of the lower problem (Px ) is not singleton. In this case, there is lack of clarity at the upper level as to which optimal solution from the lower level should be utilized. To overcome this, two problems have been studied in [5]. They are the optimistic and pessimistic problems defined respectively as follows (Pop ) min min{F(x, y)|y ∈ S(x)}
(3)
(Pp ) min max{F(x, y)|y ∈ S(x)}.
(4)
x∈X
y
and x∈X
y
The optimistic problem (Pop ) is more tractable as compared to the pessimistic problem; therefore, most of the studies handle the optimistic version of the bilevel optimization
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