OPTIMALITY CONDITIONS FOR NONSMOOTH MULTIOBJECTIVE BILEVEL OPTIMIZATION USING TANGENTIAL SUBDIFFERENTIALS

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex. Mathematics Subject Classification. 49J52, 90C46, 58E35. Received April 20, 2021. Accepted September 9, 2021.


Introduction
Bilevel programming is considered to be one of the important areas in optimization and operations research due to its many applications in various fields (economics, logistics, transportation, engineering and computer science, etc). The problem is stated at two levels of hierarchy in which the feasible set of decisions at the upper level is implicitly related with the solution set at the lower level.
This paper treats the case of bilevel programs in which the upper level decision requires minimizing a vector function : R × R → R , while the lower level involves a single objective : R × R → R. This program has the following form Keywords. Nonsmooth multiobjective optimization, bilevel programming, optimality conditions, optimal value function, constraint qualifications, tangential subdifferentials.
with : R × R → R , ∈ I, denote the lower level constraints. The same problem has been investigated by many authors. In employing both the value function as well as the KKT conditions of (1.2), necessary optimality conditions were established in [17] under smooth settings. In [4,9], necessary optimality conditions were derived for Pareto optimal solutions using the Hiriart-Urruty scalarization [7], under the use of Clarke's generalized Jacobian with the nonsmooth Mangasarian-Fromovitz constraint qualification in the first paper [4], and convexificators with a generalized Abadie's constraint qualification in the latter paper [9].
In this work, we take up the general concept of local weakly efficiency with respect to Problem (1.1) which is multiobjective. Letting be the leader's feasible region, a pair (̂︀,̂︀) ∈ is called a local weakly efficient solution of (1.1) and (1.2) if there is an open set 0 × 0 containing (̂︀,̂︀) such that Our aim is to apply the value function approach as well as the notion of tangential subdifferentials as mathematical tools in order to establish the optimality conditions that must hold at any local weakly efficient solution of the problem (1.1) and (1.2). The latter tool includes many types of subdifferentials like Gâteaux derivatives or convex subdifferentials and coincides with those of Clarke and Michel-Penot, in the case of locally Lipschitz functions that are Clarke regular and MP regular, respectively. It coincides also with upper regular convexificator in the case of tangentially convex functions. Moreover, we employ a specific constraint qualification which is weaker than the Zangwill constraint qualification [19] and other known constraint qualifications like Cottle, Mangasarian-Fromovitz, Kuhn-Tucker, etc.
After this introduction, our paper has the following structure. In Section 2, we give the notations and definitions needed in the sequel. In Section 3, we reformulate the bilevel program (1.1) and (1.2) and we present some constraint qualifications in terms of tangential subdifferentials with a statement of the relationship between them. In Section 4, necessary conditions are derived under a suitable constraint qualification. An example is given at the end to clarify the main result.

Notations and preliminaries
We follow the standard notation employed in nonlinear optimization. First, letting ∅ ̸ = ⊆ R , by the sets int , co , ∘ , we mean the interior, convex hull, negative and strictly negative polar cone of , respectively. The convex cone generated by contains exactly all conic combinations of the elements of , it can be expressed as follows: Recall that for any two sets 1 and 2 in R one has We recall also three notions of tangent cones: feasible directions, weak feasible directions and contingent, which are given with respect to and an element ∈ cl as follows It is a direct matter to check that ( , ) ⊆ ( , ) ⊆ ( , ). Note that the cone ( , ) is in general not convex nor closed, whereas ( , ) is closed but not necessarily convex, and when is convex, ( , ) = ( , ) Let : R → R. The function is said to be tangentially convex at ∈ R [15] if its directional derivative (also known as Dini derivative) at , is finite for any direction ∈ R and convex in this argument. Observe that we have sublinearity of the directional derivative of any tangentially convex function since it is positive homogeneous. Realize that if is Hadamard directionally differentiable, then its Hadamard directional derivative reduces to it directional derivative. For the converse, is Hadamard directionally differentiable at in if is locally Lipschitz at and directionally differentiable. On the other hand, the tangential subdifferential of : R → R at ∈ R is given by [10,15] For a tangentially convex function, this subdifferential is nonempty, compact and convex (see [11]). Furthermore, tangentially convex functions constitute a large class that contains convex functions on open domains where the tangential subdifferential falls into the classical Fréchet subdifferential, Gâteaux differentiable functions on open domains with a tangential subdifferential reduced to the gradient. This class also includes locally Lipschitz functions that are either Clarke regular [1] or Michel-Penot regular [13], and their tangential subdifferential is equal to that of Clarke in the first case and Michel-Penot in the second. Notice that if is tangentially convex at , then it follows from the sublinearity of where + ( , ) and − ( , ) denote respectively the upper and lower Dini directional derivatives of at in the direction . Then, ( ) is a convexificator of at . Note also that the definition of tangential subdifferential coincides with that of upper regular convexificator in the case of tangentially convex functions since ′ ( , ) = + ( , ) = sup * ∈ ( ) ⟨ * , ⟩, ∀ ∈ R .

Reformulation of the bilevel program and constraint qualifications
In order to convert the bilevel program (1.1) to an equivalent single-level program, we are going to employ the value function approach [14,18] The feasible set of (3.2) is In the following definition, we give three constraint qualifications:
Next, following [3,8] we present an upper estimate for the subdifferential of the value function (which coincides with ( ) since is convex) in terms of the initial data of (1.2).

Proposition 4.5. Suppose that
(i) is a convex and continuous function.
Then, an efficient upper estimate of the subdifferential of the value function can be determined as follows: (̂︀) ⊆ ⋃︁ Now, we present an example illustrating the main result.
Example 4.6. Consider = = 1, = = = 2 and let the leader's objectives and constraints be given by The follower's objective and constraints are as follows )︂ .
In the above example, we can obtain a similar result if we have only = 1 . In this case, it's worth noting that Theorem 5.1 of [8] and Theorem 1 of [5] cannot be employed because 1 is not locally Lipschitz at the local weakly efficient (̂︀,̂︀) = (0, 0). It follows that our main result is more general since the leader's objective can be multiobjective and not necessarily locally Lipschitz. On the other hand, since the functions 1 and 2 are not convex, we cannot apply the results that require the convexity of the functions in the upper level such as Theorem 4.1 of [2].