A note on the paper "Necessary and su¢ cient optimality conditions using convexifactors for mathematical programs with equilibrium constraints"

In this work, some counterexamples are given to refute some results in the paper by Kohli (RAIRO-Oper. Res. 53, 1617-1632, 2019). We correct the fault in some of his results. Keywords Convexifactor; Constraint qualications; Mathematical programs with equilibrium constraints; Optimality conditions. AMS Subject Classications: 90C30; 90C46; 49J52 1 Introduction Mathematical programs with equilibrium constraints have been investigated by many authors. In the paper [8], the author investigated the following mathematical programs with equilibrium constraints (MPEC) : 8><>>: Minimize f (x) s.t. ( g (x) 0; h (x) = 0; G (x) 0; H (x) 0; G (x)H (x) = 0; where f : R ! R; g : R ! R; h : R ! R; G : R ! R and H : R ! R: Under a nonsmooth constraint qualication (@ GCQ) given in terms of convexifactors, the author established rst order necessary optimality condition for (MPEC) : The main theorem, where the author gave necessary optimality conditions, is Theorem 4.4 [8]. In this article, we show that necessary optimality conditions given by Kohli in [8] are not correct. In support of our remarks, some counterexamples are given (see Example 2 and Remark 4) and some reasoning mistakes in the proof of the main result ([8, Theorem 4.4] ) are highlighted (see Remark 3, Remark 4 and Remark 7). Finally, we present corrected versions of his results. Theorem 10 is actually a corrected version of Theorem 4.4 in [8]. The rest of the paper is organized in this way: Section 2 contains basic denitions and preliminary material. Counterexamples and comments are given in Section 3. Section 4 addresses our main results (corrected optimality conditions). A conclusion is given in Section 5. 1LAMA, USMBA, FSDM, Department of Mathematics, Morocco, e-mail address : ngadhi@hotmail.com 1 This provisional PDF is the accepted version. The article should be cited as: RAIRO: RO, doi: 10.1051/ro/2021145 2 Preliminaries Throughout this section, let R be the usual n-dimensional Euclidean space. Given a nonempty subset S of R; the closure, convex hull, and convex cone (including the origin) generated by S are denoted respectively by cl S; conv S and pos S: The negative polar cone of S is dened by S := fv 2 R j hx; vi 0; 8x 2 Sg : The contingent cone T (S; x) to S at x 2 cl S is dened by T (S; x) = fv 2 R : 9tn # 0 and 9vn ! v such that x+ tnvn 2 S; 8n 2 Ng: Let f : R ! R [ f+1g be a given function and let x 2 R where f(x) is nite. The expressions f d (x; v) = lim inf t&0 [f(x+ tv) f(x)]=t and f d (x; v) = lim sup t&0 [f(x+ tv) f(x)]=t signify, respectively, the lower and upper Dini directional derivatives of f at x in the direction v. Denition 1 [2]The function f : R ! R [ f+1g is said to have an upper convexifactor @f(x) at x if @f(x) R is closed and, for each v 2 R; f d (x; v) sup x 2@uf(x) hx ; vi: The function f : R ! R[f+1g is said to have an upper semiregular convexifactor @f(x) at x if @f(x) is an upper convexifactor at x and, for each v 2 R, f d (x; v) sup x 2@usf(x) hx ; vi: 3 Counterexamples and comments The following example shows that [8, Theorem 4.4] is not correct. Example 2 Consider the optimization problem (MPEC) where f (x1; x2; x3) := x1 + x2 2x3; g (x1; x2; x3) := x3; h (x1; x2; x3) := 0; G1 (x1; x2; x3) := x1; G2 (x1; x2; x3) := x2; H1 (x1; x2; x3) := x2 and H2 (x1; x2; x3) := x1: On the one hand, the origin is the unique minimizer of (MPEC) : On the other hand, it can be seen that @f (x) := f(1; 1; 2)g is a bounded upper semiregular convexifactor of f at x := (0; 0; 0) : Moreover, @g (x) := f(0; 0; 1)g ; @h (x) := f(0; 0; 0)g ; @G1 (x) := f(1; 0; 0)g ; @ ( G1) (x) := f( 1; 0; 0)g ; @G2 (x) := f(0; 1; 0)g ; @ ( G2) (x) := f(0; 1; 0)g ; @H1 (x) := f(0; 1; 0)g ; @ ( H1) (x) := f(0; 1; 0)g ; @H2 (x) := f(1; 0; 0)g and @ ( H2) (x) := f( 1; 0; 0)g are upper convexifactors of g; h; G1; G1; G2; G2; H1; H1; H2 and H2 at x respectively. Remark that B = f1; 2g :


Introduction
Mathematical programs with equilibrium constraints have been investigated by many authors. In the paper [8], the author investigated the following mathematical programs with equilibrium constraints where : R → R, : R → R , ℎ : R → R , : R → R and : R → R . Under a nonsmooth constraint qualification ( * − ) given in terms of convexifactors, the author established first order necessary optimality condition for (MPEC). The main theorem, where the author gave necessary optimality conditions, is Theorem 4.4 of [8].
In this article, we show that necessary optimality conditions given by Kohli [8] are not correct. In support of our remarks, some counterexamples are given (see Example [8]. The rest of the paper is organized in this way: Section 2 contains basic definitions and preliminary material. Counterexamples and comments are given in Section 3. Section 4 addresses our main results (corrected optimality conditions). A conclusion is given in Section 5.

Preliminaries
Throughout this section, let R be the usual -dimensional Euclidean space. Given a nonempty subset of R , the closure, convex hull, and convex cone (including the origin) generated by are denoted respectively by , and . The negative polar cone of is defined by The contingent cone ( , ) to at ∈ is defined by The function : R → R ∪ {+∞} is said to have an upper semiregular convexifactor ( ) at if ( ) is an upper convexifactor at and, for each ∈ R ,

Counterexamples and comments
The following example shows that Theorem 4.4 of [8] is not correct.
On the one hand, the origin is the unique minimizer of (MPEC). On the other hand, it can be seen that } is a bounded upper semiregular convexifactor of at := (0, 0, 0). Moreover, We have = 2 while ≤ 1 due to (3.1) and (3.2). A contradiction.
The author did not pay attention to the cone that precedes the convex hull in the previous formula (see line-6 on page 1625). This error has seriously impacted the remaining of the proof of Theorem 4.4 from [8]. Since (3.3) is an essential part of the definition of the * -strong stationarity property, Theorem 4.4 of [8] is also not correct.
Notice that the boundedness of the sequence of the multipliers is neither acquired nor insured. The following result is a corrected version of Lemma 2.3 from [8]. Being standard, the proof has been omitted.

Optimality conditions
In the following definition, we recall the generalized alternatively stationarity concept given by Ardali et al. and is not an integral part of Definition 4.1. It is this equality that distorted Kohli's result. Notice that Remark 4.2 of [8] is not correct since condition for the sum of multipliers does not exist in [4,6,10].
Notice that if all the functions are differentiable and the upper convexifactor is replaced by the upper regular convexifactor in the above stationary notion, then this notion reduces to the A-stationary condition given by Flegel and Kanzow [6] and by Flegel [3].
We shall need the following nonsmooth constraint qualification.
where is the feasible set of (MPEC) and The following result is the corrected version of Theorem 4.4 from [8].  -Since ⊆ , we get sup -Since ( ) is also a closed set, ( ) is a compact set (see [7], Thm. 1.4.3). By Lemma 3.4, we get 0 ∈ ( ) + ( ).

Conclusions
In the paper [8], the author investigated a mathematical programs with equilibrium constraints. The main result, Theorem 4.4 of [8], and the lemma ( [8], Lem. 2.3) on which the author is based are false. In this work, counterexamples are given to refute Theorem 4.4 of [8] and Lemma 2.3 of [8]. Furthermore, we correct the flaws.