On M-stationarity conditions in MPECs and the associated qualification conditions

Abstract

Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly related to the qualification conditions on the “lower level”. For instance, even under the linear independence constraint qualification (LICQ) for a lower level feasible set described by $\mathscr {C}^1$ functions, the calmness properties of the original and the enhanced perturbation mapping are drastically different. When passing to $\mathscr {C}^{1,1}$ data, this difference still remains true under the weaker Mangasarian–Fromovitz constraint qualification, whereas under LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results are finally applied to MPECs with structured equilibria.

Appendix: A strong counterexample to the reversion of Proposition 2 under MFCQ and \(\mathscr {C}^2\) data for \(\varGamma \)

In Example 2 we have shown that under MFCQ and smooth inequalities describing the set $\varGamma$, the mapping M may be calm, whereas the enhanced mapping $\tilde{M}$ fails to be calm for some multiplier. In the following stronger counterexample we construct a set $\varGamma$ described by $\mathscr {C}^2$ inequalities satisfying MFCQ at given $\bar{y}$ and a function F such that M is calm at $(0,\bar{x},\bar{y})$ while $\tilde{M}$ is not calm at $(0,0,\bar{x},\bar{y},\lambda )$ for any $\lambda \in \varLambda (\bar{x},\bar{y})$.

Define first $\varphi _1,\varphi _2:[-1,1]\rightarrow {\mathbb {R}}$ and $q_1,q_2:[-1,1]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ as

$$\begin{aligned} \begin{aligned} \varphi _1(t)&:=\left\{ \begin{array}{ll} (-1)^k\Bigl (t-\frac{1}{k}\Bigr )^3\Bigl (t-\frac{1}{k+1}\Bigr )^3 &{} \text{ for } t\in \left[ \frac{1}{k+1},\frac{1}{k}\right] ,\ k\in {\mathbb {N}}\\ 0&{} \text{ for } t\le 0, \end{array}\right. \\ \varphi _2(t)&:=\left\{ \begin{array}{ll} (-1)^k\Bigl (t-\frac{1}{k}\Bigr )^5\Bigl (t-\frac{1}{k+1}\Bigr )^5 &{} \text{ for } t\in \left[ \frac{1}{k+1},\frac{1}{k}\right] ,\ k\in {\mathbb {N}}\\ 0&{} \text{ for } t\le 0, \end{array}\right. \\ q_1(y)&:=\varphi _1(y_1)-y_2,\\ q_2(y)&:=\varphi _2(y_1)-y_2, \end{aligned} \end{aligned}$$

put $\omega ={\mathbb {R}}$ and as the reference point take $(\bar{x},\bar{y}_1,\bar{y}_2)=(0,0,0)$. These functions are depicted in Fig. 1. Note first that MFCQ is indeed satisfied for $\varGamma$ and that $\varphi _1$ and $\varphi _2$ are twice continuously differentiable. Define further

$$\begin{aligned} \phi (t):=\max \{\varphi _1(t), \varphi _2(t)\}. \end{aligned}$$

Note that for any given k only one of functions $\varphi _1$ and $\varphi _2$ will be active in the definition of $\phi$ on interval $(\frac{1}{k+1},\frac{1}{k})$. Because $\phi '(\frac{1}{k})=\phi ''(\frac{1}{k})=0$ for all $k\in {\mathbb {N}}$, it remains to verify the twice continuous differentiability of $\phi$ at 0. But we have

$$\begin{aligned} \lim _{t\rightarrow 0}t^{-1}|\phi (t)-\phi (0)| = \lim _{t\rightarrow 0}t^{-1}|\varphi _1(t)|=0, \end{aligned}$$

which implies that $|\phi '(0)|=0$. Similarly we obtain $\phi ''(0)=0$ and that $\phi$ is twice continuously differentiable. Finally, we define $F(x,y):=(-\phi '(y_1),\ 1)$. By construction of $\phi$, we obtain that F is continuously differentiable. Since $\varGamma ={\text {epi}}\phi$ we have that

$$\begin{aligned} M(0)=\left\{ (x,y)\left| \, \begin{pmatrix}\phi '(y_1)\\ -1\end{pmatrix}\in N_\varGamma (y)\right. \right\} ={\mathbb {R}}\times {\text {gph}}\phi . \end{aligned}$$

As $M(p)\subset M(0)$ for all p small enough, we obtain that M is calm at $(0,\bar{x},\bar{y})$.

Fig. 1

Segments of graphs $\varphi _1$ and $2.3 \times 10^9\varphi _2$. The constant in front of $\varphi _2$ is used for graphical purposes

It is easy to see that $\varLambda (\bar{x},\bar{y})=\{\lambda \ge 0|\,\lambda _1+\lambda _2=1\}$. We will show now that $\tilde{M}$ is not calm at $(0,0,\bar{x},\bar{y},\lambda )$ for any $\lambda \in \varLambda (\bar{x},\bar{y})$. Define

$$\begin{aligned} \begin{aligned} \varOmega _1&:= \{t\in [0,1]|\, \varphi _1(t)=\varphi _2(t)\},\\ \varOmega _2&:= \{t\in [0,1]|\, \varphi _1(t)\ne \varphi _2(t),\ \varphi _1'(t)=\varphi _2'(t)\},\\ \varOmega _3&:= [0,1]\setminus (\varOmega _1\cup \varOmega _2) \end{aligned} \end{aligned}$$

and note that for all $t\in \varOmega _2\cup \varOmega _3$ small enough it holds that $|\varphi _2(t)|<|\varphi _1(t)|$ and for all $t\in \varOmega _3$ small enough we have $|\varphi _2'(t)|<|\varphi _1'(t)|$.

We will show first that $\hat{T}_{\{1\}}$ defined in (20) is not calm at $(0,\bar{y})$. From the definition we see that

$$\begin{aligned} \hat{T}_{\{1\}}(p)=\{y|\, \varphi _1(y_1)=y_2+p_1, \varphi _2(y_1)\le y_2+p_2\}. \end{aligned}$$

and thus

$$\begin{aligned} \hat{T}_{\{1\}}(0)=\{y|\, \varphi _1(y_1)=y_2, \varphi _2(y_1)\le y_2\}=\{(y_1,\varphi _1(y_1))|\, \varphi _1(y_1)\ge 0\}. \end{aligned}$$

Now pick any sequence $y_{k1}> 0$, $y_{k1}\rightarrow 0$ such that $y_{k1}\in \varOmega _2$ and $\varphi _1(y_{k1})<0$ and define $p_{k1}:=0$, $y_{k2}:=\varphi _1(y_{k1})$ and $p_{k2}:=\varphi _2(y_{k1})-y_{k2}$. Then $y_k\in \hat{T}_{\{1\}}(p_k)$. Moreover, as $\varphi _1$ and $\varphi _2$ have the same signs

$$\begin{aligned} 0<\Vert p_k\Vert =p_{k2}=\varphi _2(y_{k1})-y_{k2}=\varphi _2(y_{k1})-\varphi _1(y_{k1})\le |\varphi _1(y_{k1})|. \end{aligned}$$

Consider now a point $\tilde{y}_{k1}\in \varOmega _1$ at which $d(y_{k1},\varOmega _1)$ is realized. Since $\varOmega _1\subset \hat{T}_{\{1\}}(0)$ and $\varphi _1$ is zero on $\varOmega _1$, we obtain

$$\begin{aligned} \frac{|d(y_k, \hat{T}_{\{1\}}(0))|}{|p_k|}\ge \frac{|d(y_{k1}, \varOmega _1)|}{|\varphi _1(y_{k1})|}=\frac{|y_{k1}-\tilde{y}_{k1}|}{|\varphi _1(y_{k1})-\varphi _1(\tilde{y}_{k1})|}=\frac{1}{\varphi _1'(\xi _k)}, \end{aligned}$$

where in the last equality we have used the mean value theorem to find some $\xi _k$ which lies in the line segment connecting $y_{k1}$ and $\tilde{y}_{k1}$. Since $\varphi _1$ is twice continuously differentiable with $\varphi _1'(0)=0$, we have proved that $\hat{T}_{\{1\}}$ is not calm at $(0,\bar{y})$. For $\hat{T}_{\{2\}}$ we proceed with a similar construction. In this case we have

$$\begin{aligned} \hat{T}_{\{2\}}(0)=\{y|\, \varphi _1(y_1)\le y_2, \varphi _2(y_1)= y_2\}=\{(y_1,\varphi _2(y_1))|\, \varphi _1(y_1)\le 0\} \end{aligned}$$

and for the contradicting sequence we choose some $y_{k1}>0$, $y_{k1}\rightarrow 0$ such that $y_{k1}\in \varOmega _2$ and $\varphi _1(y_{k1})>0$ and define again $p_{k1}:=0$, $y_{k2}:=\varphi _1(y_{k1})$ and $p_{k2}:=\varphi _2(y_{k1})-y_{k2}$ and perform the estimates as in the previous case. Since for $\hat{T}_{\{1,2\}}$ we have

$$\begin{aligned} \hat{T}_{\{1,2\}}(0)=\{y|\, \varphi _1(y_1)= y_2, \varphi _2(y_1)= y_2\}=\{(y_1,\varphi _1(y_1))|\, \varphi _1(y_1)= 0\}, \end{aligned}$$

either of the previous contradicting sequences can be chosen.

Fix now any $\bar{\lambda }\in \varLambda (\bar{x},\bar{y})$ and consider the corresponding index set $I=\{i|\ \bar{\lambda }_i>0\}$. In the previous several paragraphs we have shown that $\hat{T}_I$ is not calm at $(0,\bar{y})$ and found a sequence $(\tilde{p}_k,\tilde{y}_k)$ violating the calmness property. By virtue of Lemma 4 we obtain that T is not calm at $(0,\bar{y},\bar{\lambda })$. Moreover, from the proof of this lemma we see that the sequence $(p_k,y_k,\lambda _k)$, which violates the calmness of T at $(0,\bar{y},\bar{\lambda })$, can be taken in such a way that $p_k=\tilde{p}_k$, $y_k=\tilde{y}_k$ and $\lambda _k=\bar{\lambda }$ with $(\tilde{y}_k,\bar{\lambda })\in T(\tilde{p}_k)$ and

$$\begin{aligned} d((\tilde{y}_k,\bar{\lambda }), T(0))> (k-1)\Vert \tilde{p}_k\Vert . \end{aligned}$$

(55)

Furthermore, in all the previous cases we have chosen $\tilde{y}_k$ in such a way that $\tilde{y}_{k1}\in \varOmega _2$.

We will show that $\tilde{M}$ is not calm at $(0,0,\bar{x},\bar{y},\bar{\lambda })$. Consider sequence

$$\begin{aligned} (0,0,\tilde{p}_{k1},\tilde{p}_{k2},\bar{x},\tilde{y}_{k1},\tilde{y}_{k2},\bar{\lambda }_1,\bar{\lambda }_2)\rightarrow (0,0,0,0,\bar{x},0,0,\bar{\lambda }_1,\bar{\lambda }_2) \end{aligned}$$

(56)

and show first that $(\bar{x},\tilde{y}_{k1},\tilde{y}_{k2},\bar{\lambda }_1,\bar{\lambda }_2)\in \tilde{M}(0,0,\tilde{p}_{k1},\tilde{p}_{k2})$, which amounts to showing

$$\begin{aligned} \begin{aligned} \begin{pmatrix}0\\ 0\end{pmatrix}&=\begin{pmatrix}-\phi '(\tilde{y}_{k1})\\ 1\end{pmatrix}+ \begin{pmatrix}\varphi _1'(\tilde{y}_{k1})&{} \varphi _2'(\tilde{y}_{k1}) \\ -1 &{}-1\end{pmatrix}\begin{pmatrix}\bar{\lambda }_1\\ \bar{\lambda }_2\end{pmatrix},\\&\quad q(\tilde{y}_k)-\tilde{p}_k\in N_{{\mathbb {R}}_+^2}(\bar{\lambda }). \end{aligned} \end{aligned}$$

We know that $(\tilde{y}_k,\bar{\lambda })\in T(\tilde{p}_k)$ and hence the inclusion is satisfied. Moreover, as $\tilde{y}_{k1}\in \varOmega _2$ by construction of this sequence and as $\bar{\lambda }_1+\bar{\lambda }_2=1$, we indeed obtain

$$\begin{aligned} (\bar{x},\tilde{y}_{k1},\tilde{y}_{k2},\bar{\lambda }_1,\bar{\lambda }_2)\in \tilde{M}(0,0,\tilde{p}_{k1},\tilde{p}_{k2}). \end{aligned}$$

(57)

From the respective definitions of $\tilde{M}$ and T, we infer that $\tilde{M}(0,0,0,0)\subset {\mathbb {R}}^n\times T(0,0)$ and consequently due to (55) we obtain

$$\begin{aligned} d((\bar{x},\tilde{y}_{k1},\tilde{y}_{k2},\bar{\lambda }_1,\bar{\lambda }_2), \tilde{M}(0,0,0,0))\ge & {} d((\tilde{y}_{k1},\tilde{y}_{k2},\bar{\lambda }_1,\bar{\lambda }_2), T(0,0))\\> & {} (k-1)\Vert \tilde{p}_k\Vert . \end{aligned}$$

This together with (56) and (57) implies that $\tilde{M}$ is indeed not calm at $(0,0,\bar{x},\bar{y},\bar{\lambda })$. Since $\bar{\lambda }$ was chosen arbitrarily from $\varLambda (\bar{x},\bar{y})$, the construction has been completed.