On quasidifferentiable mathematical programs with equilibrium constraints

The aim of this article is to study mathematical programs with equilibrium constraints involving quasidifferentiable functions, denoted by QMPEC, and to synthesize suitable optimality conditions. We first derive Fritz-John (FJ) necessary optimality conditions with Lagrange multipliers depending upon the choice of superdifferentials. We introduce a suitable variant of no nonzero abnormal multiplier constraint qualification for the QMPEC, denoted by NNAMCQ-QMPEC, and derive Karush–Kuhn–Tucker (KKT) necessary optimality conditions. We also propose some conditions under which the Lagrange multipliers do not depend upon the choice of superdifferentials. Further, we prove several sufficient optimality conditions for a weak stationary point to be optimal for the QMPEC under suitable choice of generalized convex functions.


Introduction
Mathematical programs with equilibrium constraints (MPECs), which is an extension of bilevel optimization, is utilised extensively in a variety of fields (Luo et al. 1996;Dempe 2003;Colson et al. 2007).Standard Fritz-John (FJ) conditions were applied by Flegel and Kanzow (2003) to obtain new stationary conceptions for MPECs.In order to construct M-stationarity conditions for the MPECs, a constraint qualification (CQ) of the Abadie type was created in Flegel and Kanzow (2005b); Ye (2005) and its relationships with other CQs were examined.Strong stationarity has been demonstrated in Flegel and Kanzow (2005b) to be a crucial optimality requirement under Guignard CQ (GCQ) in the setting of MPECs.Flegel et al. (2007) used optimality conditions of disjunctive programming to solve MPECs.A (iii) x ≦ y ⇔ x i ≤ y i , ∀ i ∈ P; (iv) x ≤ y ⇔ x i ≤ y i , ∀ i ∈ P and x ≠ y.
The sequel will make use of the following fundamental ideas in nonsmooth analysis.
Definition 1 Let Υ be a nonempty subset of the Euclidean space ℝ p .A func- tion h ∶ ℝ p → ℝ is locally Lipschitz continuous at  ∈ Υ ⊂ ℝ p with Lipschitz constant L h iff there exist a nonnegative scalar L h and  > 0 such that, for all x 1 , x 2 ∈ ( ; ) ∩ Υ , one has The function h is Lipschitz continuous with Lipschitz constant L h on Υ iff for all x 1 , x 2 ∈ Υ , one has Definition 2 (Clarke 1983) Let Υ be a nonempty subset of ℝ p and let h ∶ ℝ p → ℝ be locally Lipschitz at ∈ Υ .The Clarke directional derivative of h at in a direction v ∈ ℝ p , denoted by h • ( ;v) , is given by and the Clarke subdifferential of h at , denoted by • h( ) , is given by where t ↓ 0 means t → 0 + .
Definition 3 A function h ∶ ℝ p → ℝ is directionally differentiable at ∈ ℝ p in a direction v ∈ ℝ p iff the limit exists and is finite.We say that h is directionally differentiable or semi-differentiable at iff its directional derivative h � ( ;v) exists and is finite for every v ∈ ℝ p .

Remark 2
The function h is uniformly directionally differentiable at , if the convergence in above definition holds uniformly wrt any unit vector v. Further, it was shown in Demyanov and Rubinov (1995) that the directional differentiability is equivalent to the uniformly directional differentiability for a locally Lipschitz function.

|h(x
Definition 4 (Demyanov and Rubinov 2000) A real-valued function h ∶ ℝ p → ℝ is quasidifferentiable at ∈ ℝ p iff h is directionally differentiable at and there exists a pair of convex compact sets h() ⊂ ℝ p and h() ⊂ ℝ p such that where h( ) and h( ) are called subdifferential and superdifferential of h at , respectively.Further, the pair of sets D h ( ) ∶= [ h( ), h( )] is called quasidiffer- ential of the function h at .

Remark 3
The choice of h( ) and h( ) is also a quasidifferential of h at for any compact set C ⊂ ℝ p .Moreover, when h is differentiable, a natural choice of subdifferential and superdifferential is {∇h( )} and {0}, respectively.Some of the important properties of quasidifferentiable functions are listed below: Proposition 1 (Demyanov and Rubinov 2000) Let two real-valued functions h 1 and h 2 on ℝ n be quasidifferentiable at a point ∈ ℝ n with quasidifferentials , respectively, and let ∈ ℝ .Then, the following statements are true: (a) The function h ∶= h 1 + h 2 is also quasidifferentiable at with quasidifferential (b) The function h ∶= h 1 is also quasidifferentiable at with quasidifferential Gao (2000a) studied the following scalar optimization problem (P) with mixed constraints: where and f , g j (j ∈ M) and h k (k ∈ N) are quasidifferentiable real-valued functions on ℝ p .
x ∈ Υ P , (P) A point ∈ Υ P is a minimizer of P iff f ( ) ≤ f (x) for every x ∈ Υ P .A point ∈ Υ P is a local minimizer of P iff there exists an open ball ( , ) at with radius  > 0 such that f ( ) ≤ f (x) for every x ∈ Υ P ∩ ( , ).The collection of all mini- mizers and local minimizers of P are denoted by Υ m P and Υ lm P , respectively.Gao (2000a) obtained the following FJ optimality conditions for P with variable scalar multipliers.
Theorem 1 (Gao 2000a, Theorem 3.4) Let ∈ Υ m P such that f , g j (j ∈ M) are quasidifferentiable at and h k (k ∈ N) are locally Lipschitz in a neighbourhood of .Then, for any sets of * f ∈ f ( ), * g j ∈ g j ( ) (j ∈ M), there exist scalars f ( ) ≥ 0, g j ( ) ≥ 0 (j ∈ M) and h k ( ) (k ∈ N), not all zero, such that where f ( ), g j ( ) (j ∈ M) and h k ( ) (k ∈ N) depends on the specific choice of ).
Further, Gao (2000a) obtained FJ conditions for P with fixed multipliers under the following two hypothesis: (H1) For any x ∈ ℝ p , one has where and (H2) For every v ∈ Φ(x), the sets and

Necessary optimality conditions
Consider a mathematical program with inequality, equality, and equilibrium constraints as follows: where where It is easy to see that Υ P( ) is contained in Υ QMPEC (see, e.g., Flegel and Kanzow 2005a).

Variable Lagrange multipliers
Now, we are ready to prove a necessary optimality condition for QMPEC as follows: Theorem 3 (Fritz-John necessary optimality for QMPEC) , therefore by Theorem 1, for any set of min f (x) s. t. x ∈ Υ P( ) , (P( )) (1) , not all zero, such that and which implies by the property of quasidifferentials Proposition 1 and by the property of Clarke subdifferentials (see, e.g.Clarke 1983) that, for any set of , not all zero, such that and , depends on the specific choice of which implies by the definition of index sets the required result.◻ If all the involved functions in the QMPEC are locally Lipschitz, then we have the following corollary.
Corollary 1 (Fritz-John conditions under Lipschitz continuity) and Proof The proof is a direct consequence of the fact that for a locally Lipschitz directionally differentiable function, the superdifferential is singleton zero and the subdifferential is equal to the Clarke subdifferential.◻ To prove a Karush-Kuhn-Tucker type necessary optimality condition, we will assume that no nonzero abnormal multipliers exist (see, e.g., Ye 2000).
say that a no nonzero abnormal multiplier constraint qualification for QMPEC, denoted by NNAMCQ-QMPEC, is satisfied at iff 30 Page 10 of 20 implies Based on the above constraint qualification, we may derive the following KKT condition.
Theorem 4 (Karush-Kuhn-Tucker necessary optimality for QMPEC) Proof By the assumptions in the theorem, by Theorem 3, for any set of , there exist such that (1) and ( 2) are satisfied.Suppose to the contrary that βf () = 0.Then, by NNAMCQ-QMPEC at , we get all the multipliers as zero, which is a contradiction to the conclusion of Theorem 3 and hence βf () > 0. Now, by defining , and  H ∶= βH βf , we get the required multipliers.◻ (5)

Fixed Lagrange multipliers
Now, for a given point ∈ ℝ p , we denote and Now, we are ready to prove necessary optimality conditions under stronger assumptions.

Theorem 5 (Necessary optimality conditions)
Then, there exist ∶= ( f , g , h , G , H ) ∈ ℝ 1+m+n+2 l ⧵{0} such that and Moreover, if we further assume that there is no nonzero abnormal multiplier at , that is, Moreover, f , g j (j ∈ M), h k (k ∈ N), H i (i ∈ L) and G i (i ∈ L) are uniformly directionally dif- ferentiable at with Ω QMPEC ( ) = Φ QMPEC ( ) .Then, by Theorem 2, there exist sca- , not all zero, such that and which implies by the definition of index sets that Further, suppose to the contrary that f = 0.Then, (11) holds which implies (12), a contradiction to the fact that not all the Lagrange multipliers are zero and hence the result.◻ The following numerical example validate the outputs of the above theorem.
• The directional derivatives of f , H 1 and G 1 at along any direction v ∶= (v 1 , v 2 ) ∈ ℝ 2 are given by respectively.
• The quasidifferentials of f , H 1 and G 1 at are given by and respectively, where which gives • Moreover, • Now, H 1 = 0 = G 1 and hence NNAMCQ-QMPEC is satisfied at .• Then, by above theorem there exist f = 1, H 1 = 1 and G 1 = 1 2 such that

Sufficient optimality conditions
We can give the following stationary points for the QMPEC based on the different stationary points in the literature of MPECs (see, e.g.Ye 2005).

Definition 6 (Stationary points for
• a W-stationary point of the QMPEC iff there exist ∶= ( g , h , G , H ) ∈ ℝ m+n+2 l such that and If in addition to ( 14) and ( 15), one has Remark 4 If the involved functions of the QMPEC are all continuously differentiable, then the stationary conditions for the QMPEC reduce to different stationarity conditions of MPEC (Ye 2005).It is easy to observe that the S−stationary condition is strongest and the W−stationary condition is the weakest among all the stationary conditions.
We need the following indices to prove that the above stationary point is a sufficient condition for optimality under certain assumptions.
Based on the notion of F−convex functions wrt convex compact set given by Antc- zak (Antczak 2016, Definition 4.2) and generalized (F, )−convexity given by Nobakhtian and Nobakhtian (2006), Singh and Laha (2022a) gave the following notions of generalized F− convexity in terms of convex compact set.

Definition 7
Let h be a real-valued quasidifferentiable function on ℝ p with qua- sidifferential sublinear in the third argument, and let S h ( ) ∶= h( ) + h( ) be a convex com- pact set.The function h is (i) F−quasiconvex at ∈ ℝ p over ℝ p wrt S h ( ) iff for any x ∈ ℝ p , one has (ii) F−pseudoconvex at ∈ ℝ p over ℝ p wrt S h ( ) iff for any x ∈ ℝ p , one has Remark 5 If a function is quasidifferentiable F−convex in the sense of Antczak (2016), then it is both quasidifferentiable F−quasiconvex and F−pseudoconvex wrt the same convex compact set.For continuously differentiable functions, the definitions reduce to F−quasiconvexity and F−pseudoconvexity introduced by Hanson and Mond (1986).For locally Lipschitz functions, the definitions reduce to (F, 0)− quasiconvexity and (F, 0)−pseudoconvexity by Nobakhtian (Nobakhtian and Nobakhtian 2006, Definition 2.2).
The following theorem gives a sufficient condition for a weak stationary point of the QMPEC to be optimal.