Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications

Abstract

In this paper, we introduce two new constraint qualifications for mathematical programs with equilibrium constraints. One of them is a relaxed version of the No Nonzero Abnormal Multiplier Constraint Qualification, and the other is an adaptation of the Constant Rank of Subspace Component. The new conditions have nice properties. Indeed, they have the local preservation property and imply the error bound property under mild assumptions. Thus, they can be used to extend some known results on stability and sensitivity analysis. Furthermore, they can also be used in the convergence analysis of several methods for solving mathematical programs with equilibrium constraints.

Appendix: Proofs

Proof of Lemma 3.2

The proof is by induction on the number of complementary constraints. Our assumption is that, for every constraint system $\varOmega$ with m complementary constraints such that MPEC-CRSC holds at some feasible point $x^{*}$, there is a $\delta >0$ (which may depend on $\varOmega$ and $x^{*}$) such that for each feasible point $y \in \varOmega \cap \mathbb {B}(x^{*}, \delta )$, the set $\mathcal {G}(x;A_{-}(y), \mathcal {E}, \mathcal {I}(y), \mathcal {K}(y))$ has the same rank for every x near y.

When $m=0$ MPEC-CRSC coincides with CRSC and since CRSC has a local preservation property (Lemma 5.4 of [10]), the conclusion of the lemma is valid for $m=0$. Now, assume that the lemma holds whenever the number of complementary constraints is less than $m_{0}$. Assume that $m=m_{0}$. Now, we will analyze the following cases:

• 1. Case$\mathcal {J}(x^{*})$is an empty set. Consider the following constraint system

  $$\begin{aligned}&\widehat{\varOmega }:= \left\{ x \in \mathbb {R}^{n}: g(x)\le 0, h(x)=0, H_{i}(x)=0, G_{j}(x)=0, \right. \\&\quad \left. i \in \mathcal {I}(x^{*}), j \in \mathcal {K}(x^{*}) \right\} . \end{aligned}$$

  Since MPEC-CRSC holds for $\varOmega$ at $x^{*}$, CRSC holds for $\widehat{\varOmega }$ at $x^{*}$. By Theorem 5.4 of [10], there is a $\delta _1>0$ such that $\mathcal {G}(x;A_{-}(y, \widehat{\varOmega }), \mathcal {E}, \mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$ has the same rank for every x near y, $\forall y \in \widehat{\varOmega } \cap \mathbb {B}(x^{*}, \delta )$ and $A_{-}(y, \widehat{\varOmega })=A_{-}(x^{*}, \widehat{\varOmega })$, for every $y \in \widehat{\varOmega } \cap \mathbb {B}(x^{*}, \delta )$. Take $\delta _{2}>0$ such that $\mathcal {I}(x^{*})\subset \mathcal {I}(y)$, $\mathcal {K}(x^{*})\subset \mathcal {K}(y)$ and $\mathcal {J}(y)\subset \mathcal {J}(x^{*})$, $\forall y \in \varOmega \cap \mathbb {B}(x^{*},\delta _2)$. Clearly, $\mathcal {I}(y)\setminus \mathcal {I}(x^*) \subset \mathcal {J}(x^{*})$ and $\mathcal {K}(y)\setminus \mathcal {K}(x^*) \subset \mathcal {J}(x^{*})$. As $\mathcal {J}(x^{*})=\emptyset$, $\mathcal {I}(x^{*})=\mathcal {I}(y)$ and $\mathcal {K}(x^{*})=\mathcal {K}(y)$. Since $A(y,\widehat{\varOmega })=A(y)$, we get that $A_{-}(y,\widehat{\varOmega })=A_{-}(y)$, for every y in $\varOmega \cap \mathbb {B}(x^{*},\delta _2)$. Thus, $\mathcal {G}(x;A_{-}(y, \widehat{\varOmega }), \mathcal {E}, \mathcal {I}(x^{*}), \mathcal {K}(x^{*}))= \mathcal {G}(x;A_{-}(y), \mathcal {E}, \mathcal {I}(y), \mathcal {K}(y))$ has the same rank for every x near y, for every $y \in \varOmega \cap \mathbb {B}(x^{*},\min \{\delta _{1},\delta _{2}\})$.

• 2. Case$\mathcal {J}(x^{*})$is a non-empty set. Assume, by contradiction, that there is a sequence $\{y^{k}\}\subset \varOmega$ with $y^{k} \rightarrow x^{*}$ such that $\mathcal {G}(x;A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$ does not have the same rank for some x near $y^{k}$. Without loss of generality (possibility after taken an adequate subsequence), we can assume that $\mathcal {I}(x^{*})\subset \mathcal {I}(y^{k})$, $\mathcal {K}(x^{*})\subset \mathcal {K}(y^{k})$$\mathcal {J}(y^{k})\subset \mathcal {J}(x^{*})$ and $A_{-}(x^{*})\subset A_{-}(y^{k})$ (Lemma 3.1), $\forall k \in \mathbb {N}$. Furthermore, we can suppose that $\mathcal {I}(y^{k})$, $\mathcal {K}(y^{k})$, $\mathcal {J}(y^{k})$ and $A_{-}(y^{k})$ do not depend on k, namely, $\mathcal {I}$, $\mathcal {K}$, $\mathcal {J}$ and $A_{-}$, respectively. Now, depending whether $\mathcal {J}$ is an empty set or not, we have the following sub-cases:

  • (a) If $\mathcal {J}$ is not an empty set. Here, consider the constraint system

    $$\begin{aligned} \widetilde{\varOmega _{1}}:= \left\{ x \in \mathbb {R}^{n}: \begin{array}{ll} &{}g(x)\le 0, h(x)=0, \\ &{}0 \le H_{i}(x) \bot G_{i}(x) \ge 0, i \in \{1,\dots ,m_{0}\} \setminus \mathcal {J} \end{array} \right\} . \end{aligned}$$

    (30)

    Here, MPEC-CRSC holds at $x^{*}$ for $\widetilde{\varOmega }_{1}$ . Indeed, we see that $\mathcal {I}(x^{*}, \widetilde{\varOmega }_{1})=\mathcal {I}(x^{*})$, $\mathcal {K}(x^{*}, \widetilde{\varOmega }_{1})=\mathcal {K}(x^{*})$, $A(x^{*}, \widetilde{\varOmega _{1}})=A(x^{*})$ and $A_{-}(x^{*}, \widetilde{\varOmega }_{1})=A_{-}(x^{*})$. As $\widetilde{\varOmega }_{1}$ has less complementary constraints than $\varOmega$ we conclude, by induction, that there is a $\delta _{1}>0$ such that for every $y \in \widetilde{\varOmega }_{1} \cap \mathbb {B}(x^{*}, \delta _{1})$, $\mathcal {G}(x;A_{-}(y,\widetilde{\varOmega }_{1} ), \mathcal {E}, \mathcal {I}(y,\widetilde{\varOmega }_{1}),\mathcal {K}(y, \widetilde{\varOmega _{1}}))$ has the same rank for every x near y. In particular, for $y=y^{k} \in \varOmega$ for k large enough. But, since $\mathcal {J}=\mathcal {J}(y^{k})$, we get $\mathcal {I}(y^{k},\widetilde{\varOmega }_{1})= \mathcal {I}(y^{k}) \cap (\{1,\dots ,m_{0}\} \setminus \mathcal {J})=\mathcal {I}(y^{k})$, $\mathcal {K}(y^{k},\widetilde{\varOmega }_{1})= \mathcal {K}(y^{k}) \cap (\{1,\dots ,m_{0}\} \setminus \mathcal {J})=\mathcal {K}(y^{k})$ and hence $A_{-}(y^{k},\widetilde{\varOmega _{1}} )=A_{-}(y^{k})$. Thus, $\mathcal {G}(x;A_{-}(y^{k},\widetilde{\varOmega }_{1}), \mathcal {E}, \mathcal {I}(y^{k},\widetilde{\varOmega _{1}}),\mathcal {K}(y^{k}, \widetilde{\varOmega _{1}}))= \mathcal {G}(x;A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}),\mathcal {K}(y^{k}))$ has the same rank for every x near $y^{k}$, which is a contradiction.

  • (b) If $\mathcal {J}$ is an empty set. Define the constraint system

    $$\begin{aligned} \widetilde{\varOmega }_{2}:=\left\{ x \in \mathbb {R}^{n} : \begin{array}{lll} &{} g_{j}(x)\le 0, j \in A_{+}(x^{*}), \\ &{} g_{j}(x)=0, j \in A_{-}(x^{*}), h_{i}(x)=0, i \in \mathcal {E} \\ &{} 0 \le H_{i}(x) \bot G_{i}(x) \ge 0 \text { for } i \in \{1,\dots ,m_{0}\} \end{array} \right\} . \end{aligned}$$

    (31)

    Note that MPEC-RCPLD holds on $\widetilde{\varOmega _{2}}$ at $x^{*}$. Thus, by Theorem 4.3 of [24], MPEC-RCPLD holds on $\widetilde{\varOmega _{2}}$ for every $y \in \widetilde{\varOmega _{2}}$ close to $x^{*}$. In particular, for $y=y^{k} \in \varOmega$ for k large enough due to $A_{-}(x^{*}) \subset A_{-}(y^{k})$. Now, since $\mathcal {J}(y^{k})=\mathcal {J}=\emptyset$, MPEC-RCPLD holds on $\widetilde{\varOmega _{2}}$ at $y^{k}$ and hence RCPLD holds for $\widetilde{\varOmega }_{3}$ at $y^{k}$, where

    $$\begin{aligned} \widetilde{\varOmega }_{3}:=\left\{ x \in \mathbb {R}^{n} : \begin{array}{lll} &{} g_{j}(x)\le 0, j \in A_{+}(x^{*}), g_{j}(x)=0, j \in A_{-}(x^{*}) \\ &{} h(x)=0, H_{i}(x)=0 \text { for } i \in \mathcal {I}, G_{j}(x)=0 \text { for } j \in \mathcal {K} \end{array} \right\} . \end{aligned}$$

    (32)

    By Theorem 4.3 of [10], RCPLD implies CRSC. From CRSC, we conclude that $\mathcal {G}(x;A_{-}(y^{k},\widetilde{\varOmega }_{3}) \cup A_{-}(x^{*}), \mathcal {E}, \mathcal {I},\mathcal {K})$ has the same rank for every x near $y^{k}$. Now, we will show that $A_{-}(y^{k},\widetilde{\varOmega }_{3})=A_{-}(y^{k})\cap A_{+}(x^{*})$. Take $j \in A_{-}(y^{k})\cap A_{+}(x^{*})$, then $-\nabla g_{j}(y^{k}) \in \text {span}_{+}\mathcal {G}(y^{k}; A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$. We observe that the cone $\text {span}_{+} \mathcal {G}(y^{k}; A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$ is a subset of the sum of the linear subspace $\text {span}\{\nabla g_{i}(y^{k}):i \in A_{-}(x^{*})\}$ and $\text {span}_{+}\mathcal {G}(y^{k}; A_{-}(y^{k})\setminus A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$. Such sum of sets is in $\text {span}_{+}\mathcal {G}(y^{k};A(y^{k},\widetilde{\varOmega }_{3}), A_{-}(x^{*})\cup \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$ and thus we get that $j \in A_{-}(y^{k},\widetilde{\varOmega }_{3})$. Now, we will prove the other inclusion. Take j in $A_{-}(y^{k},\widetilde{\varOmega }_{3})$. By (32), we have that $j \in A_{+}(x^{*})$ and that $-\nabla g_{j}(y^{k})$ belongs to $\text {span}_{+}\mathcal {G}(y^{k}; A_{-}(y^{k})\cap A_{+}(x^{*}),\mathcal {E},\mathcal {I}(y^{k}),\mathcal {K}(y^{k}))+ \text {span}\{\nabla g_{i}(y^{k}):i \in A_{-}(x^{*})\}$. But, this set is included into $\text {span}_{+}\mathcal {G}(y^{k}; A_{-}(y^{k}),\mathcal {E},\mathcal {I}(y^{k}),\mathcal {K}(y^{k}))$ because $A_{-}(x^{*})$ is a subset of $A_{-}(y^{k})$. Thus, we conclude that $j \in A_{-}(y^{k})\cap A_{+}(x^{*})$.

    Now, from $A_{-}(y^{k},\widetilde{\varOmega }_{3})=A_{-}(y^{k})\cap A_{+}(x^{*})$, $\mathcal {I}(y^{k})=\mathcal {I}$ and $\mathcal {K}(y^{k})=\mathcal {K}$ we have that $\mathcal {G}(x;A_{-}(y^{k},\widetilde{\varOmega }_{3}) \cup A_{-}(x^{*}), \mathcal {E},\mathcal {I},\mathcal {K})= \mathcal {G}(x;[A_{-}(y^{k})\cap A_{+}(x^{*})] \cup A_{-}(x^{*}), \mathcal {E},\mathcal {I},\mathcal {K})$ has the same rank for every x near $y^{k}$. But, the last set is $\mathcal {G}(x;A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}),\mathcal {K}(y^{k}))$. Thus, we obtain a contradiction with the choice of the sequence $\{y^{k}\}$.

In all the cases, there is a $\delta >0$ such that, for each feasible point $y \in \varOmega \cap \mathbb {B}(x^{*}, \delta )$, $\mathcal {G}(x;A_{-}(y), \mathcal {E}, \mathcal {I}(y), \mathcal {K}(y))$ has the same rank for every x near y. $\square$

Proof of Theorem 3.1

Assume that MPEC-CRSC holds at $x^{*}$. We will show that MPEC-CRSC holds for every feasible point near to $x^{*}$. By Lemma 3.2, (H1) holds for every feasible point close to $x^{*}$. Thus, we will show that (H2) holds for every $y \in \varOmega$ close to $x^{*}$.

Now assume, by contradiction, that there is a feasible sequence $\{y^{k}\}\subset \varOmega$ with $y^{k} \rightarrow x^{*}$ such that (H2) does not hold at $y^{k}$. For each $k\in \mathbb {N}$, take index subsets $A_{-}^{'}(y^{k})\subset A_{-}(y^{k})$, $\mathcal {E}^{'}(y^{k}) \subset \mathcal {E}$, $\mathcal {I}^{'}(y^{k}) \subset \mathcal {I}(y^{k})$ and $\mathcal {K}^{'}(y^{k}) \subset \mathcal {K}(y^{k})$ such that $\mathcal {G}(y^{k};A_{-}^{'}(y^{k})), \mathcal {E}^{'}, \mathcal {I}^{'}(y^{k}), \mathcal {K}^{'}(y^{k}))$ is a basis for $\text {span } \mathcal {G}(y^{k};A_{-}(y^{k})), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$ and $\mathcal {G}(y^{k};\emptyset , \mathcal {E}^{'}, \mathcal {I}^{'}(y^{k}), \mathcal {K}^{'}(y^{k}))$ is a basis for $\text {span } \mathcal {G}(y^{k};\emptyset , \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$. Since MPEC-CRSC does not hold at $y^{k}$. There are sets $A^{'}_{+}(y^{k}) \subset A_{+}(y^{k})$, $\mathcal {J}_{H}^{'}(y^{k}), \mathcal {J}_{G}^{'}(y^{k}) \subset \mathcal {J}(y^{k})$, and multipliers $(\lambda ^{k}, \mu ^{k}, u^{k}, v^{k})$ which are not all zero with $\text {supp}(\mu ) \subset A_{-}^{'}(y^{k})\cup A_{+}^{'}(y^{k})$, $\text {supp}(\lambda ) \subset \mathcal {E}^{'}(y^{k})$, $\text {supp}(u) \subset \mathcal {I}^{'}(y^{k})\cup \mathcal {J}_{H}^{'}(y^{k})$, $\text {supp}(v) \subset \mathcal {K}^{'}(y^{k})\cup \mathcal {J}_{G}^{'}(y^{k})$, $\mu _{j}^{k}\in \mathbb {R}_{+}$, $(j \in A_{+}^{'}(y^{k}))$ and either $u^{k}_{\ell }v^{k}_{\ell }=0$ or $u^{k}_{\ell }>0, v^{k}_{\ell }>0$ for each $\ell \in \mathcal {J}(y^{k})$ such that

$$\begin{aligned} \sum _{j=1}^{p} \mu _{j}^{k}\nabla g_{j}(y^{k})+ \sum _{i=1}^{q} \lambda ^{k}_{i}\nabla h_{i}(y^{k})- \sum _{\imath =1}^{m} u^{k}_{\imath }\nabla H_{\imath }(y^{k})- \sum _{\jmath =1}^{m} v^{k}_{\jmath }\nabla G_{\jmath }(y^{k})=0, \end{aligned}$$

(33)

and for every positive sequence $\delta ^{k}$ with $\delta ^{k} \rightarrow 0$, there is a point $x^{k}$ in $\mathbb {B}(y^{k}, \delta ^{k})$ such that $\mathcal {G}(x^k;A_{+}^{'}(y^{k})\cup A_{-}^{'}(y^{k}), \mathcal {E}, \mathcal {I}^{'}(y^{k})\cup \mathcal {J}_{H}^{'}(y^{k}), \mathcal {K}^{'}(y^{k})\cup \mathcal {J}_{G}^{'}(y^{k}))$ is linearly independent.

Now, define the vector $\omega ^{k}$ as

$$\begin{aligned} \sum _{j \in A_{-}^{'}(y^{k})} \mu _{j}^{k}\nabla g_{j}(y^{k})+ \sum _{i \in \mathcal {E}^{'}(y^{k})} \lambda ^{k}_{i}\nabla h_{i}(y^{k})- \sum _{\imath \in \mathcal {I}^{'}(y^{k})} u^{k}_{\imath }\nabla H_{\imath }(y^{k})- \sum _{\jmath \in \mathcal {K}^{'}(y^{k})} v^{k}_{\jmath }\nabla G_{\jmath }(y^{k}). \end{aligned}$$

(34)

By definition of $A_{-}(y^{k})$, we see that $\omega ^{k}$ belongs to $\text {span}_{+}\mathcal {G}(y^{k}, A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$ (see Remark 3.2). Then, $\omega ^{k}$ can be written as

$$\begin{aligned} \omega ^{k}= & {} \sum _{j \in A_{+}^{''}(y^{k})} \hat{\mu }_{j}^{k}\nabla g_{j}(y^{k})+ \sum _{i \in \mathcal {E}^{'}(y^{k})} \hat{\lambda }^{k}_{i}\nabla h_{i}(y^{k})\nonumber \\&- \sum _{\imath \in \mathcal {I}^{'}(y^{k})} \hat{u}^{k}_{\imath }\nabla H_{\imath }(y^{k})- \sum _{\jmath \in \mathcal {K}^{'}(y^{k})} \hat{v}^{k}_{\jmath }\nabla G_{\jmath }(y^{k}), \end{aligned}$$

(35)

where $A_{+}^{''}(y^{k})\subset A_{-}(y^{k})$ and $\hat{\mu }_{j}^{k} \ge 0$, for $j \in A_{+}^{''}(y^{k})$. By Lemma 2.2, we can take $A_{+}^{''}(y^{k})$ such that $\mathcal {G}(y^{k};A_{+}^{''}(y^{k}), \mathcal {E}^{'}(y^{k}), \mathcal {I}^{'}(y^{k}), \mathcal {K}^{'}(y^{k}))$ is a linearly independent set. Set

$$\begin{aligned} \begin{array}{llll} &{} \tilde{\lambda }_{i}^{k}:= \hat{\lambda }_{j}^{k} (i \in \mathcal {E}^{'}(y^{k})), \ \ &{}\tilde{\mu }_{j}^{k}:= \mu _{j}^{k} (j \in A_{+}^{'}(y^{k})), \ \ \tilde{\mu }_{j}^{k}:= \hat{\mu }_{j}^{k} (j \in A_{+}^{''}(y^{k})),\\ &{} &{}\tilde{u}_{j}^{k}:= u_{j}^{k} (j \in \mathcal {J}_{H}^{'}(y^{k})), \ \ \tilde{u}_{j}^{k}:= \hat{\mu }_{j}^{k} (j \in \mathcal {I}^{'}(y^{k})),\\ &{} &{} \tilde{v}_{j}^{k}:= v_{j}^{k} (j \in \mathcal {J}_{G}^{'}(y^{k})), \ \ \tilde{v}_{j}^{k}:= \hat{v}_{j}^{k} (j \in \mathcal {K}^{'}(y^{k})) \ \ \ \ \end{array} \end{aligned}$$

(36)

with $\text {supp}(\tilde{\mu }^k) \subset A_{-}^{''}(y^{k})\cup A_{+}^{'}(y^{k})$, $\text {supp}(\tilde{\lambda }^k) \subset \mathcal {E}^{'}(y^{k})$, $\text {supp}(\tilde{u}^k) \subset \mathcal {I}^{'}(y^{k})\cup \mathcal {J}_{H}^{'}(y^{k})$ and $\text {supp}(\tilde{v}^k) \subset \mathcal {K}^{'}(y^{k})\cup \mathcal {J}_{G}^{'}(y^{k})$. Now, substituting (35) into (33), we get

$$\begin{aligned} \sum _{j=1}^{p} \tilde{\mu }_{j}^{k}\nabla g_{j}(y^{k})+ \sum _{i=1}^{q} \tilde{\lambda }^{k}_{i}\nabla h_{i}(y^{k})- \sum _{\imath =1}^{m} \tilde{u}^{k}_{\imath }\nabla H_{\imath }(y^{k})- \sum _{\jmath =1}^{m} \tilde{v}^{k}_{\jmath }\nabla G_{\jmath }(y^{k})=0. \end{aligned}$$

(37)

After dividing (37) by $\Vert (\tilde{\lambda }^{k}, \tilde{\mu }^{k}, \tilde{u}^{k}, \tilde{v}^{k})\Vert$, we assume that $\Vert (\tilde{\lambda }^{k}, \tilde{\mu }^{k}, \tilde{u}^{k}, \tilde{v}^{k})\Vert =1$.

Since the set of indexes is finite, we assume without loss of generality (possibly after taking an adequate subsequence) that $\{A_{+}^{'}(y^{k}), A_{+}^{''}(y^{k}), A_{-}^{'}(y^{k}), \mathcal {E}^{'}(y^{k}), \mathcal {I}^{'}(y^{k}), \mathcal {J}_{H}^{'}(y^{k})$, $\mathcal {K}^{'}(y^{k})$, $\mathcal {J}_{G}^{'}(y^{k})\}$ does depend on k, namely $\{A_{+}^{'}, A_{+}^{''}, A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {J}_{H}^{'}, \mathcal {K}^{'}, \mathcal {J}_{G}^{'}\}$. It is not difficult to see that $\tilde{\gamma }^{k}:=(\tilde{\mu }^{k}, \tilde{\lambda }^{k}, (\tilde{u}_{1}^{k}, \tilde{v}_{1}^{k}), \dots , (\tilde{u}_{m}^{k}, \tilde{v}_{m}^{k})) \in N_{\varLambda }(F(y^{k}))$, where F is given by (3). Since $\Vert \tilde{\gamma }^{k}\Vert =1$, we assume that $\tilde{\gamma }^{k}\rightarrow \hat{\gamma }$ and $\Vert \tilde{\gamma }\Vert =1$. By the outer semi-continuity of normal cone, we get that $\tilde{\gamma } \in N_{\varLambda }(F(x^{*}))$. Denote $\tilde{\gamma }$ by $(\tilde{\mu }, \tilde{\lambda }, (\tilde{u}_{1}, \tilde{v}_{1}), \dots , (\tilde{u}_{m}, \tilde{v}_{m}))$. Clearly, $\tilde{\mu }\ge 0$ and either $\tilde{u}_{\ell }\tilde{v}_{\ell }=0$ or $\tilde{u}_{\ell }>0, \tilde{v}_{\ell }>0$, $\forall \ell \in \mathcal {J}(x^{*})$. Now, taking limit in (37) we get

$$\begin{aligned}&\sum _{j \in A_{+}^{'}\cup A_{+}^{''}} \tilde{\mu }_{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \tilde{\lambda }_{i}\nabla h_{i}(x^{*})- \sum _{\imath \in \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}} \tilde{u}_{\imath }\nabla H_{\imath }(x^{*})\nonumber \\&\quad - \sum _{\jmath \in \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'}} \tilde{v}_{\jmath }\nabla G_{\jmath }(x^{*})=0. \end{aligned}$$

(38)

By MPEC-CRSC, the set $\mathcal {G}(x;A_{+}^{'}\cup A_{+}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is linearly dependent for every x near $x^{*}$. But, since (i) $\mathcal {G}(x;A_{+}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a linearly independent set for every x such that $\Vert x-y^{k}\Vert \le \delta _{1}^{k}$ (for some $\delta _{1}^{k}>0$ and k large enough) and (ii) $\text {span } \mathcal {G}(x;A_{+}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a subspace of $\text {span } \mathcal {G}(x;A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$, where $\mathcal {G}(x;A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a linear basis for $\mathcal {G}(x;A_{-}(y^{k}), \mathcal {E}, \mathcal {I}(y^{k}), \mathcal {K}(y^{k}))$ for every x such that $\Vert x-y^{k}\Vert \le \delta _{2}^{k}$ for some $\delta ^{k}_{2}>0$ (this follows from the inclusion $A_{+}^{''} \subset A_{-}(y^{k})$ and Lemma 3.2). We conclude that the set $\mathcal {G}(x;A_{+}^{'}\cup A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is linearly dependent for every vector x such that $\Vert x-y^{k}\Vert \le \min \{\delta _{1}^{k},\delta _{2}^{k}\}$ for k sufficiently large. Now, define $\delta ^{k}:=\min \{\delta _{1}^{k},\delta _{2}^{k},1/k\}$. Clearly, we have that $\delta ^{k}\rightarrow 0$. Note that for every x with $\Vert x-y^{k}\Vert \le \delta ^{k}\le \min \{\delta _{1}^{k},\delta _{2}^{k}\}$, the set $\mathcal {G}(x;A_{+}^{'}\cup A_{+}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is linearly dependent, which is a contradiction. $\square$

Proof of Theorem 3.2

By Lemma 3.2, we only need to prove that (H3) holds for every $y \in \varOmega$ near to $x^*$. Suppose by contradiction that RNNAMCQ fails for some sequence $y^k \in \varOmega$ with $y^k \rightarrow x^*$. Using a similar same reasoning as in the proof of the above theorem, we obtain multipliers $\tilde{\gamma }:=(\tilde{\mu }, \tilde{\lambda }, (\tilde{u}_{1}, \tilde{v}_{1}), \dots , (\tilde{u}_{m}, \tilde{v}_{m}))$ with $\tilde{\gamma } \in N_{\varLambda }(F(x^*))$ and $\Vert \tilde{\gamma }\Vert =1$ such that

$$\begin{aligned}&\sum _{j \in A_{+}^{'}\cup A_{+}^{''}} \tilde{\mu }_{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \tilde{\lambda }_{i}\nabla h_{i}(x^{*})- \sum _{\imath \in \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}} \tilde{u}_{\imath }\nabla H_{\imath }(x^{*})\\&\quad - \sum _{\jmath \in \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'}} \tilde{v}_{\jmath }\nabla G_{\jmath }(x^{*})=0, \end{aligned}$$

which is a contradiction with RNNAMCQ. $\square$

Proof of Theorem 4.1

We will use induction. The hypothesis is that for every constraint system with m complementary constraints, MPEC-RCPLD implies MPEC-CRSC. Clearly, the theorem is true when $m=0$ since RCPLD implies CRSC. Now, assume that MPEC-RCPLD implies MPEC-CRSC whenever the number of complementary constraints is less than $m_{0}$. To show that MPEC-RCPLD implies MPEC-CRSC when the number of complementary constraints is $m_{0}$, we consider the following cases:

• 1. Case$\mathcal {J}(x^{*})$is an empty set. Consider the following constraint system

  $$\begin{aligned} \varOmega _{1}:=\left\{ x \in \mathbb {R}^{n} : \begin{array}{lll} &{} g_{j}(x)\le 0, h(x)=0 \\ &{} H_{i}(x)=0, G_{j}(x)=0 \text { for } i \in \mathcal {I}(x^{*}), j \in \mathcal {K}(x^{*}) \end{array} \right\} . \end{aligned}$$

  (39)

  Since MPEC-RCPLD holds for $\varOmega$ at $x^{*}$, RCPLD and CRSC hold for $\varOmega _{1}$ at $x^{*}$. But, if $\varOmega _{1}$ conforms CRSC at $x^*$ then $\varOmega$ holds MPEC-CRSC at $x^{*}$.

• 2. Case$\mathcal {J}(x^{*})$is a non-empty set. Now, consider the next constraint system

  $$\begin{aligned} \varOmega _{2}:=\left\{ x \in \mathbb {R}^{n} : \begin{array}{lll} &{} g_{j}(x)\le 0, h(x)=0 \\ &{} 0 \le H_{i}(x) \bot G_{i}(x) \ge 0 \text { for } i \in \{1,\dots ,m_{0}\} \setminus \mathcal {J}(x^{*}) \end{array} \right\} . \end{aligned}$$

  (40)

  Hence MPEC-RCPLD holds at $x^{*}$ for $\varOmega$, MPEC-RCPLD holds for $\varOmega _2$ at $x^{*}$. But, as $\varOmega _2$ has less complementary constraints than $\varOmega$, by induction we have that MPEC-CRSC holds for $\varOmega _2$ at $x^{*}$. Thus, the set $\mathcal {G}(y;A_{-}(x^{*}, \varOmega _{2}), \mathcal {E}, \mathcal {I}(x^{*},\varOmega _2), \mathcal {K}(x^{*},\varOmega _2))$ has the same rank for y near $x^*$. But, since $\mathcal {I}(x^{*},\varOmega _{2})=\mathcal {I}(x^{*})$, $\mathcal {K}(x^{*},\varOmega _{2})=\mathcal {K}(x^{*})$ and $A_{-}(x^{*}, \varOmega _{2})=A_{-}(x^{*})$, the set $\mathcal {G}(y;A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$ has the same rank for y sufficiently close $x^{*}$. Thus, (H1) holds.

  Now, take four sets of indexes $A_{-}^{'} \subset A_{-}(x^{*})$, $\mathcal {E}^{'} \subset \mathcal {E}$, $\mathcal {I}^{'} \subset \mathcal {I}(x^{*})$ and $\mathcal {I}^{'} \subset \mathcal {K}(x^{*})$ such that $\mathcal {G}(x^{*};A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'},\mathcal {K}^{'})$ is a basis for $\text {span } \mathcal {G}(x^*;A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$. Moreover, we can assume that the set $\mathcal {G}(x^{*};\emptyset ,\mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a linear basis for the subspace $\text {span } \mathcal {G}(x^{*};\emptyset , \mathcal {E}, \mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$. In order to complete the proof of the theorem, we will show that for each $A^{'}_{+} \subset A_{+}(x^{*})$ and $\mathcal {J}_{H}^{'}, \mathcal {J}_{G}^{'} \subset \mathcal {J}(x^{*})$, if there are multipliers $(\lambda , \mu , u, v)\in \mathbb {R}^{p}\times \mathbb {R}^{q}\times \mathbb {R}^{m}\times \mathbb {R}^{m}$ which are not all zero with $\mu _{j}\in \mathbb {R}_{+} ( \forall j \in A_{+}^{'})$ and either $u_{\ell }v_{\ell }=0$ or $u_{\ell }>0, v_{\ell }>0$ for each $\ell \in \mathcal {J}(x^{*})$ such that

  $$\begin{aligned}&\sum _{j \in A_{+}^{'}\cup A_{-}^{'}} \mu _{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \lambda _{i}\nabla h_{i}(x^{*})+ \sum _{\imath \in \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}} u_{\imath }\nabla H_{\imath }(x^{*})\nonumber \\&\quad + \sum _{\jmath \in \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'}} v_{\jmath }\nabla G_{\jmath }(x^{*})=0. \end{aligned}$$

  (41)

  Then, $\mathcal {G}(y;A_{+}^{'}\cup A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is a linearly dependent set for all $y \in \mathbb {B}(x^{*}, \delta )$. Assume that (41) holds for some multipliers $\{\lambda , \mu , u, v\}$, not all zero, such that $\mu _{j}\in \mathbb {R}_{+}$, $j \in A_{+}^{'}$ and either $u_{\ell }v_{\ell }=0$ or $u_{\ell }>0, v_{\ell }>0$ for each $\ell \in \mathcal {J}(x^{*})$. Denote

  $$\begin{aligned} \omega :=\sum _{j \in A_{-}^{'}} \mu _{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \lambda _{i}\nabla h_{i}(x^{*})+ \sum _{\imath \in \mathcal {I}^{'}} u_{\imath }\nabla H_{\imath }(x^{*})+ \sum _{\jmath \in \mathcal {K}^{'}} v_{\jmath }\nabla G_{\jmath }(x^{*}). \end{aligned}$$

  (42)

  As $\text {span}_{+}\mathcal {G}(x^{*}; A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}), \mathcal {K}(x^{*}))= \text {span} \mathcal {G}(x^{*}; A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}), \mathcal {K}(x^{*}))$ and by using Remark 3.2, we see that $\omega \in \text {span}_{+} \mathcal {G}(x^{*}; A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}), \mathcal {K}(x^{*}))$. Thus, there are a subset $A_{-}^{''} \subset A_{-}(x^{*})$ and a multiplier $\hat{\mu } \in \mathbb {R}_{+}^{q}$ with $\text {supp}(\hat{\mu })\subset A_{-}^{''}$ such that

  $$\begin{aligned} \omega =\sum _{j \in A_{-}^{''}} \hat{\mu }_{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \lambda _{i}\nabla h_{i}(x^{*})+ \sum _{\imath \in \mathcal {I}^{'}} u_{\imath }\nabla H_{\imath }(x^{*})+ \sum _{\jmath \in \mathcal {K}^{'}} v_{\jmath }\nabla G_{\jmath }(x^{*}). \end{aligned}$$

  (43)

  By Lemma 2.2, we can assume that $\mathcal {G}(x^{*};A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a linearly independent set and so $\mathcal {G}(y;A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ for every $y \in \mathbb {B}(x^{*},\delta _{2})$ for some $\delta _{2}>0$. Substituting (43) into (41) and using MPEC-RCPLD, we get that $\mathcal {G}(y;A_{+}^{'}\cup A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is a linearly dependent set for every y near $x^{*}$.

  In resume, we have proven that: (i) every y close to $x^{*}$ the set $\mathcal {G}(y;A_{+}^{'}\cup A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is linearly dependent; (ii) that $\mathcal {G}(y;A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a linearly independent set for y close to $x^{*}$ and that (iii) $\text {span }\mathcal {G}(y;A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a subset of $\text {span }\mathcal {G}(y;A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'},\mathcal {K}^{'})$ for every y close to $x^{*}$. From these results, we have that $\mathcal {G}(y;A_{+}^{'}\cup A_{-}^{''}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {J}_{G}^{'})$ is a linearly dependent set for every y near $x^{*}$.

In all the mentioned cases, we can conclude that MPEC-CRSC is valid at $x^{*}$. $\square$

Proof of Theorem 4.2

Let $\omega ^{*}$ be an element of $\limsup _{(x,z)\rightarrow (x^*,F(x^{*})) } \nabla F(x)^{\top }N_{\varLambda }(z)$. Thus, there are sequences $\{x^{k}, z^{k}, \omega ^{k}, \gamma ^{k}\}$ such that $x^{k} \rightarrow x^{*}$, $z^{k} \rightarrow F(x^{*})$ and $\omega ^{k} \rightarrow \omega ^*$, where $\omega ^{k}:=\nabla F(x^{k})^{\top }\gamma ^{k}$ and $\gamma ^{k} \in N_{\varLambda }(z^{k})$. By Remark 2 of [14], we can assume that $\gamma ^{k} \in \widehat{N}_{\varLambda }(z^{k})$. Denote $\gamma ^{k}$ by $(\mu ^{k}, \lambda ^{k},(u_1^{k}, v_1^{k}),\dots ,(u_m^{k},v_m^{k}))$. Thus, for k large enough, we have that

$$\begin{aligned} \omega ^{k}=\sum _{j=1}^{p} \mu ^{k}_{j}\nabla g_{j}(x^{k}) +\sum _{i=1}^{q} \lambda ^{k}_{i}\nabla h_{i}(x^{k}) -\sum _{\imath =1}^{m} u^{k}_{\imath }\nabla H_{\imath }(x^{k}) -\sum _{\jmath =1}^{m} v^{k}_{\jmath }\nabla G_{\jmath }(x^{k}), \end{aligned}$$

(44)

for some multipliers $(\mu ^{k},\lambda ^{k},u^{k},v^{k}) \in \mathbb {R}_{+}^{p}\times \mathbb {R}^{q} \times \mathbb {R}^{m}\times \mathbb {R}^{m}$ such that $\text {supp}(\mu ) \subset A(x^{*})$, $\text {supp}(\lambda ) \subset \mathcal {E}$, $\text {supp}(u) \subset \mathcal {I}(z^{k})\cup \mathcal {J}(z^{k})$, $\text {supp}(v) \subset \mathcal {K}(z^{k})\cup \mathcal {J}(z^{k})$ and $u^{k}_{\ell }\ge 0, v^{k}_{\ell }\ge 0$, for all $\ell \in \mathcal {J}(z^{k})$. Besides, we can assume that $\mathcal {I}(x^{*})\subset \mathcal {I}(z^{k})$, $\mathcal {K}(x^{*})\subset \mathcal {K}(z^{k})$ and $\mathcal {J}(z^{k})\subset \mathcal {J}(x^{*})$, for all $k \in \mathbb {N}$. Decompose $\omega ^{k}$ as $\omega _{+}^{k}+\omega _{-}^{k}$, where

$$\begin{aligned} \omega _{-}^{k}:=\sum _{j \in A_{-}(x^{*})} \mu ^{k}_{j}\nabla g_{j}(x^{k}) +\sum _{i \in \mathcal {E}} \lambda ^{k}_{i}\nabla h_{i}(x^{k}) -\sum _{\imath \in \mathcal {I}(x^{*})} u^{k}_{\imath }\nabla H_{\imath }(x^{k}) -\sum _{\jmath \in \mathcal {K}(x^{*})} v^{k}_{\jmath }\nabla G_{\jmath }(x^{k}) \end{aligned}$$

(45)

and $\omega _{+}^{k}:=\omega ^{k}-\omega _{-}^{k}$.

Now, take sets of indexes $A_{-}^{'} \subset A_{-}(x^{*})$, $\mathcal {E}^{'} \subset \mathcal {E}$, $\mathcal {I}^{'} \subset \mathcal {I}(x^{*})$ and $\mathcal {I}^{'} \subset \mathcal {K}(x^{*})$ such that $\mathcal {G}(x^{*};A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a basis of $\text {span } \mathcal {G}(x^{*};A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$. By MPEC-CRSC, the set $\mathcal {G}(x; A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}, \mathcal {K}^{'})$ is a basis for $\text {span } \mathcal {G}(x;A_{-}(x^{*}), \mathcal {E}, \mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$ for every x near $x^{*}$, in particular for $x=x^{k}$. Thus, we write $\omega _{-}^{k}$ as

$$\begin{aligned} \omega _{-}^{k}=\sum _{j \in A_{-}^{'}} \hat{\mu }^{k}_{j}\nabla g_{j}(x^{k}) +\sum _{i \in \mathcal {E}^{'}} \hat{\lambda }^{k}_{i}\nabla h_{i}(x^{k}) -\sum _{\imath \in \mathcal {I}^{'}} \hat{u}^{k}_{\imath }\nabla H_{\imath }(x^{k}) -\sum _{\jmath \in \mathcal {K}^{'}} \hat{v}^{k}_{\jmath }\nabla G_{\jmath }(x^{k}), \end{aligned}$$

(46)

for some multipliers $(\hat{\mu }^k, \hat{\lambda }^k, \hat{u}^k,\hat{v}^k) \in \mathbb {R}^{p}\times \mathbb {R}^{q}\times \mathbb {R}^{m}\times \mathbb {R}^{m}$ such that $\text {supp}(\hat{\mu }^k)\subset A_{-}^{'}$, $\text {supp}(\hat{\lambda }^{k})\subset \mathcal {E}^{'}$, $\text {supp}(\hat{u}^{k})\subset \mathcal {I}^{'}$ and $\text {supp}(\hat{v}^{k})\subset \mathcal {K}^{'}$. For a given $k \in \mathbb {N}$, using Lemma 2.2, we can find subset of indexes $A_{+}^{'}(k) \subset A_{+}(x^{*})$, $\mathcal {I}^{'}_{+}(k) \subset \mathcal {I}(x^{k})\setminus \mathcal {I}(x^{*})$, $\mathcal {K}^{'}_{+}(k) \subset \mathcal {K}(x^{k})\setminus \mathcal {K}(x^{*})$ and $\mathcal {J}_{H}^{'}(k), \mathcal {J}_{G}^{'}(k) \subset \mathcal {J}(x^{k})$ such that

$$\begin{aligned} \omega _{+}^{k}=\sum _{j \in A_{+}^{'}(k)} \tilde{\mu }^{k}_{j}\nabla g_{j}(x^{k}) -\sum _{\imath \in \mathcal {I}^{'}_{+}(k)\cup \mathcal {J}_{H}^{'}(k)} \tilde{u}^{k}_{\imath }\nabla H_{\imath }(x^{k}) -\sum _{\jmath \in \mathcal {K}^{'}_{+}(k)\cup \mathcal {J}_{G}^{'}(k)} \tilde{v}^{k}_{\jmath }\nabla G_{\jmath }(x^{k}), \end{aligned}$$

(47)

for $(\tilde{\mu }^k, \tilde{u}^k,\tilde{v}^k)\in \mathbb {R}_{+}^{p}\times \mathbb {R}^{m}\times \mathbb {R}^{m}$ with $\text {supp}(\tilde{\mu }^k)\subset A_{+}^{'}(k)$, $\text {supp}(\tilde{u}^{k})\subset \mathcal {I}^{'}_{+}(k)\cup \mathcal {J}_{H}^{'}(k)$, $\text {supp}(\tilde{v}^{k})\subset \mathcal {K}^{'}_{+}(k)\cup \mathcal {J}_{G}^{'}(k)$, $\tilde{u}^{k}_{\ell }, \tilde{v}^{k}_{\ell }\ge 0$, $\ell \in \mathcal {J}(z^{k})$ and the following set of vectors $\mathcal {G}(x^{k};A_{+}^{'}(k)\cup A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {I}^{'}_{+}(k)\cup \mathcal {J}_{H}^{'}(k), \mathcal {K}^{'}\cup \mathcal {K}^{'}_{+}(k)\cup \mathcal {J}_{H}^{'}(k))$ is linearly independent. Since there are only a finite number of possible subset of indexes, we can assume, after taken an adequate subsequence, that $A_{+}^{'}(k)$, $\mathcal {I}^{'}_{+}(k)$, $\mathcal {K}^{'}_{+}(k)$, $\mathcal {J}_{H}^{'}(k)$ and $\mathcal {J}_{G}^{'}(k)$ do not depend on $k \in \mathbb {N}$. Denote them by $A_{+}^{'}$, $\mathcal {I}^{'}_{+}$, $\mathcal {K}^{'}_{+}$, $\mathcal {J}_{H}^{'}$ and $\mathcal {J}_{G}^{'}$, respectively. Thus, $\omega ^k$ can be written as

$$\begin{aligned}&\sum _{j \in A_{+}^{'}\cup A_{-}^{'}} \bar{\mu }^{k}_{j}\nabla g_{j}(x^{k}) +\sum _{i \in \mathcal {E}^{'}} \bar{\lambda }^{k}_{i}\nabla h_{i}(x^{k}) -\sum _{\imath \in \mathcal {I}^{'}\cup \mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}} \bar{u}^{k}_{\imath }\nabla H_{\imath }(x^{k})\nonumber \\&\quad -\sum _{\jmath \in \mathcal {K}^{'}\cup \mathcal {K}^{'}_{+}\cup \mathcal {J}_{G}^{'}} \bar{v}^{k}_{\jmath }\nabla G_{\jmath }(x^{k}), \end{aligned}$$

(48)

where the multipliers $(\bar{\mu }^k, \bar{\lambda }^k, \bar{u}^k,\bar{v}^k) \in \mathbb {R}^{p}\times \mathbb {R}^{q}\times \mathbb {R}^{m}\times \mathbb {R}^{m}$ are given by

$$\begin{aligned} \begin{array}{llll} &{}\bar{\lambda }^{k}_{i}:=\hat{\lambda }^{k}_{i}(i \in \mathcal {E}^{'}), &{}\bar{\mu }^{k}_{j}:=\tilde{\mu }^{k}_{j} (j \in A_{+}^{'}), &{}\bar{\mu }^{k}_{j}:=\hat{\mu }^{k}_{j} (j \in A_{-}^{'}) \\ &{} &{}\bar{u}^{k}_{\imath }:=\hat{u}_{\imath }^{k}(\imath \in \mathcal {I}^{'}), &{}\bar{u}^{k}_{\imath }:=\tilde{u}^{k}_{\imath } (\imath \in \mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}) \\ &{} &{} \bar{v}^{k}_{\jmath }:=\hat{v}_{\jmath }^{k}(\jmath \in \mathcal {K}^{'}), &{}\bar{v}^{k}_{\jmath }:=\tilde{v}^{k}_{\jmath }(\jmath \in \mathcal {K}^{'}_{+}\cup \mathcal {J}_{G}^{'})) \end{array} \end{aligned}$$

(49)

with $\text {supp}(\bar{\mu }) \subset A_{-}^{'}\cup A_{+}^{'}$, $\text {supp}(\bar{\lambda }) \subset \mathcal {E}^{'}$, $\text {supp}(\bar{u}) \subset \mathcal {I}^{''}\cup \mathcal {J}_{H}^{'}$ and $\text {supp}(\bar{v}) \subset \mathcal {K}^{''}\cup \mathcal {J}_{G}^{'}$. It is not difficult to see, rearranging, that

$$\begin{aligned} \bar{\gamma }^{k}:= (\bar{\mu }^{k}, \bar{\lambda }^{k}, (\bar{u}_1^{k}, \bar{v}_1^{k}),\dots ,(\bar{u}_m^{k}, \bar{v}_m^{k})) \text { belongs to } N_{\bar{\varLambda }}(\bar{z}^{k}), \end{aligned}$$

(50)

where $\bar{\varLambda }:=\mathbb {R}_{-}^{p-|A_(x^*)|}\times \{0\}^{|\mathcal {E}|+|A_{-}(x^*)|}\times \mathcal {C}^{m}$, $\bar{z}^{k}_{i}:=z_{i}^{k}$, $\forall i \notin A_{-}(x^{*})$ and $\bar{z}_{i}^{k}=0$ otherwise. Now, the sequence $\{(\bar{\mu }^{k}, \bar{\lambda }^{k}, \bar{u}^{k}, \bar{v}^{k})\}$ has a bounded subsequence; otherwise, dividing expression (48) by $M_{k}:=\Vert (\bar{\mu }^{k}, \bar{\lambda }^{k}, \bar{u}^{k}, \bar{v}^{k})\Vert$ and taking an adequate convergent subsequence of $M_{k}^{-1}(\bar{\mu }^{k}, \bar{\lambda }^{k}, \bar{u}^{k}, \bar{v}^{k})$, says $\{(\bar{\mu }, \bar{\lambda }, \bar{u}, \bar{v})\}$, we obtain that

$$\begin{aligned}&\sum _{j \in A_{+}^{'}\cup A_{-}^{'}} \bar{\mu }_{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \bar{\lambda }_{i}\nabla h_{i}(x^{*}) -\sum _{\imath \in \mathcal {I}^{'}\cup \mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}} \bar{u}_{\imath }\nabla H_{\imath }(x^{*}) \nonumber \\&\quad -\sum _{\jmath \in \mathcal {K}^{'}\cup \mathcal {K}^{'}_{+}\cup \mathcal {J}_{G}^{'}} \bar{v}_{\jmath }\nabla G_{\jmath }(x^{*}) \end{aligned}$$

(51)

is equal to zero, where $(\bar{\mu }, \bar{\lambda }, \bar{u}, \bar{v}))\ne 0$. From the outer semi-continuity of the normal cone $N_{\bar{\varLambda }}$, we get that $(\bar{\mu }, \bar{\lambda }, (\bar{u}_1, \bar{v}_1),\dots ,(\bar{u}_m, \bar{v}_m)) \in N_{\bar{\varLambda }}(F(x^*))$. Thus, we conclude that $\bar{\mu }_{j}\ge 0$$(j \in A^{'}_{+})$ and either $\bar{u}_{\ell }\bar{v}_{\ell }=0$ or $\bar{u}_{\ell }>0, \bar{v}_{\ell }>0$ for each $\ell \in \mathcal {J}(x^{*})$.

Since (51) is equal to zero with multipliers satisfying the hypothesis (H2), by MPEC-CRSC, we conclude that the set $\mathcal {G}(y;A_{+}^{'}\cup A_{-}^{'}, \mathcal {E}^{'}, \mathcal {I}^{'}\cup \mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}, \mathcal {K}^{'}\cup \mathcal {K}^{'}_{+}\cup \mathcal {J}_{H}^{'})$ is linearly dependent for every y near $x^{*}$, in particular for $y=x^{k}$, which is a contradiction.

Thus, $\{(\bar{\mu }^{k}, \bar{\lambda }^{k}, \bar{u}^{k}, \bar{v}^{k})\}$ has a convergent subsequence and without loss of generality, we assume that $(\bar{\mu }^{k}, \bar{\lambda }^{k}, \bar{u}^{k}, \bar{v}^{k})$ itself converges to $(\bar{\mu }, \bar{\lambda }, \bar{u}, \bar{v})$. By (50) and the outer semi-continuity of the normal cone, we get that $(\bar{\mu }, \bar{\lambda }, (\bar{u}_1, \bar{v}_1),\dots ,(\bar{u}_m, \bar{v}_m)) \in N_{\bar{\varLambda }}(F(x^*))$ and $\omega ^{*}$ can be written as

$$\begin{aligned}&\sum _{j \in A_{+}^{'}\cup A_{-}^{'}} \bar{\mu }_{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \bar{\lambda }_{i}\nabla h_{i}(x^{*}) -\sum _{\imath \in \mathcal {I}^{'}\cup \mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}} \bar{u}_{\imath }\nabla H_{\imath }(x^{*}) \nonumber \\&\quad -\sum _{\jmath \in \mathcal {K}^{'}\cup \mathcal {K}^{'}_{+}\cup \mathcal {J}_{G}^{'}} \bar{v}_{\jmath }\nabla G_{\jmath }(x^{*}), \end{aligned}$$

(52)

where $\bar{\mu }_{j}\in \mathbb {R}_{+}$, ($j \in A_{+}^{'}$), $\bar{\mu }_{j}\in \mathbb {R}$, ($j \in A_{-}^{'}$), either $\bar{u}_{\ell }\bar{v}_{\ell }=0$ or $\bar{u}_{\ell }>0, \bar{v}_{\ell }>0$ ($\forall \ell \in \mathcal {J}(x^{*})$), $\text {supp}(\bar{\mu })\subset A_{+}^{'}\cup A_{-}^{'}$, $\text {supp}(\bar{\lambda })\subset \mathcal {E}^{'}$$\text {supp}(\bar{u})\subset \mathcal {I}^{'}\cup \mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}$ and $\text {supp}(\bar{v})\subset \mathcal {K}^{'}\cup \mathcal {K}^{'}_{+}\cup \mathcal {J}_{G}^{'}$. But, since $\mathcal {I}(x^{*})$, $\mathcal {K}(x^{*})$ and $\mathcal {J}(x^{*})$ is a partition of $\{1,\dots , m\}$, we have that

$$\begin{aligned} \sum _{j \in A_{-}^{'}} \bar{\mu }_{j}\nabla g_{j}(x^{*})+ \sum _{i \in \mathcal {E}^{'}} \bar{\lambda }_{i}\nabla h_{i}(x^{*})+ \sum _{\imath \in \mathcal {I}^{'}} \bar{u}_{\imath }\nabla H_{\imath }(x^{*})+ \sum _{\jmath \in \mathcal {K}^{'}} \bar{v}_{\jmath }\nabla G_{\jmath }(x^{*}) \end{aligned}$$

(53)

belongs to $\text {span}_{+} \mathcal {G}(x^{*};A_{-}(x^{*}),\mathcal {E},\mathcal {I}(x^{*}),\mathcal {K}(x^{*}))$. Furthermore, noting that $A_{+}^{'}\subset A_{+}(x^{*})$, $\mathcal {I}^{'}_{+}\cup \mathcal {J}_{H}^{'}\subset \mathcal {J}(x^{*})$, $\mathcal {K}^{'}_{+}\cup \mathcal {J}_{G}^{'} \subset \mathcal {J}(x^{*})$ with $\bar{\mu }_{j}\ge 0$ ($j \in A_{+}^{'}$), we get from (53) and from (52) that $\omega ^{*}$ is an element of $\nabla F(x^{*})^{\top }N_{\varLambda }(F(x^{*}))$ as we wanted to prove. $\square$