First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

Abstract

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.

Appendix

Proof of Proposition 2.6

Firstly, we will show that (16) holds for the case that $A=\Lambda (A)$. For any $H\in \mathcal{S}^{n}$, denote $Y:=A+H$. Let $P\in \mathcal{O}^{n}$ (depending on $H$) be such that

$$\begin{aligned} \Lambda (A)+H=P\Lambda (Y)P^{T}. \end{aligned}$$

(57)

Let $\delta >0$ be any fixed number such that $0<\delta <\frac{\lambda _{|\alpha |}}{2}$ if $\alpha \ne \emptyset$ and be any fixed positive number otherwise. Then, define the following continuous scalar function

$$\begin{aligned} f (t):=\left\{ \begin{array}{l@{\quad }l@{\quad }l} t &{} \mathrm{if} &{} t>\delta ,\\ 2t-\delta &{} \mathrm{if} &{} \frac{\delta }{2} < t< \delta , \\ 0 &{} \mathrm{if} &{}t< \frac{\delta }{2}. \end{array} \right. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \{\lambda _{1}(A), \ldots , \lambda _{|\alpha |}(A)\}\in (\delta ,+\infty )\quad \mathrm{and} \quad \{\lambda _{|\alpha |+1}(A),\ldots , \lambda _{n}(A) \}\in \left( -\infty , \frac{\delta }{2}\right) . \end{aligned}$$

For the scalar function $f$, let $F:\mathcal{S}^{n}\rightarrow \mathcal{S}^{n}$ be the corresponding Löwner’s operator [25], i.e., for any $Z\in \mathcal{S}^{n}$,

$$\begin{aligned} F(Z):= \sum _{i=1}^n f(\lambda _{i}(Z)) u_i u_i^T, \end{aligned}$$

(58)

where $U\in \mathcal{O}^{n}$ satisfies that $Z=U\Lambda (Z)U^{T}$. Since $f$ is real analytic on the open set $(-\infty , \frac{\delta }{2})\cup (\delta ,+\infty )$, we know from [52, Theorem 3.1] that $F$ is analytic at $A$. Therefore, since $A=\Lambda (A)$, it is well-known (see e.g., [4, Theorem V.3.3]) that for $H$ sufficiently close to zero,

$$\begin{aligned} F(A+H)-F(A)-F'(A)H=O(\Vert H\Vert ^{2}) \end{aligned}$$

(59)

and

$$\begin{aligned} F'(A)H=\left[ \begin{array}{l@{\quad }l@{\quad }l} H_ {\alpha \alpha } &{} H_{\alpha \beta } &{} \Sigma _{\alpha \gamma } \circ H_{\alpha \gamma }\\ H_{\alpha \beta }^T &{} 0&{} 0 \\ {\Sigma }_{\alpha \gamma }^T \circ H_{\alpha \gamma }^T &{} 0 &{} 0 \end{array} \right] , \end{aligned}$$

where $\Sigma \in \mathcal{S}^{n}$ is given by (15) . Let $R(\cdot ):=\Pi _{\mathcal{S}_{+}^{n}}(\cdot )-F(\cdot )$. By the definition of $f$, we know that $F(A) =A_{+}:=\Pi _{\mathcal{S}_{+}^{n}}(A)$, which implies that $R(A)=0$. Meanwhile, it is clear that the matrix valued function $R$ is directionally differentiable at $A$, and from (14), the directional derivative of $R$ for any given direction $H\in \mathcal{S}^{n}$, is given by

$$\begin{aligned} R' (A;H)=\Pi '_{\mathcal{S}_{+}^{n}}(A;H)-F '(A)H=\left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0\\ 0&{} \Pi _{\mathcal{S}_{+}^{|\beta |}}(H_{\beta \beta }) &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right] . \end{aligned}$$

(60)

By the Lipschitz continuity of $\lambda (\cdot )$, we know that for $H$ sufficiently close to zero,

$$\begin{aligned} \{\lambda _{1}(Y), \ldots , \lambda _{|\alpha |}(Y) \}\in (\delta ,+\infty ), \quad \{\lambda _{|\alpha |+1}(Y),\ldots , \lambda _{|\beta |}(Y) \}\in \left( -\infty , \frac{\delta }{2}\right) \end{aligned}$$

and

$$\begin{aligned} \{\lambda _{|\beta |+1}(Y),\ldots , \lambda _{n}(Y) \}\in (-\infty , 0). \end{aligned}$$

Therefore, by the definition of $F$, we know that for $H$ sufficiently close to zero,

$$\begin{aligned} R(A+H)=\Pi _{\mathcal{S}_{+}^{n}}(A+H)-F (A+H) = P\left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0 \\ 0 &{} (\Lambda (Y)_{\beta \beta })_{+} &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right] P^{T}. \end{aligned}$$

(61)

Since $P$ satisfies (57), we know that for any $\mathcal{S}^{n}\ni H\rightarrow 0$, there exists an orthogonal matrix $Q\in \mathcal{O}^{|\beta |}$ such that

$$\begin{aligned} P_{\beta }=\left[ \begin{array}{l} O(\Vert H\Vert ) \\ P_{\beta \beta } \\ O(\Vert H\Vert ) \end{array} \right] \quad \mathrm{and}\quad P_{\beta \beta }=Q+O(\Vert H\Vert ^{2}), \end{aligned}$$

(62)

which was stated in [51] and was essentially proved in the derivation of Lemma 4.12 in [50]. Therefore, by noting that $(\Lambda (Y)_{\beta \beta })_{+}=O(\Vert H\Vert )$, we obtain from (60), (61) and (62) that

$$\begin{aligned}&R(A+H)-R(A)-R'(A;H)\\&\quad = \left[ \begin{array}{l@{\quad }l@{\quad }l} O(\Vert H\Vert ^{3}) &{} O(\Vert H\Vert ^{2}) &{} O(\Vert H\Vert ^{3})\\ O(\Vert H\Vert ^{2}) &{} P_{\beta \beta }( \Lambda (Y)_{\beta \beta })_{+}P_{\beta \beta }^{T}-\Pi _{\mathcal{S}_{+}^{|\beta |}}(H_{\beta \beta }) &{} O(\Vert H\Vert ^{2}) \\ O(\Vert H\Vert ^{3}) &{} O(\Vert H\Vert ^{2}) &{} O(\Vert H\Vert ^{3}) \end{array} \right] \\&\quad = \left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0\\ 0 &{} Q( \Lambda (Y)_{\beta \beta })_{+}Q^{T}-\Pi _{\mathcal{S}_{+}^{|\beta |}}(H_{\beta \beta })&{}0\\ 0 &{} 0 &{} 0 \end{array} \right] +O(\Vert H\Vert ^{2}). \end{aligned}$$

By (57) and (62), we know that

$$\begin{aligned} \Lambda (Y)_{\beta \beta }\!=\!P_{\beta }^{T}\Lambda (A)P_{\beta }\!+\!P_{\beta }^{T}HP_{\beta }\!=\! P_{\beta \beta }^{T}H_{\beta \beta }P_{\beta \beta }\!+\!O(\Vert H\Vert ^{2})\!=\! Q^{T}H_{\beta \beta }Q\!+\!O(\Vert H\Vert ^{2})\!. \end{aligned}$$

Since $Q\in \mathcal{O}^{|\beta |}$, we have

$$\begin{aligned} H_{\beta \beta }=Q\Lambda (Y)_{\beta \beta }Q^{T}+O(\Vert H\Vert ^{2}). \end{aligned}$$

By noting that $\Pi _{\mathcal{S}_{+}^{|\beta |}}(\cdot )$ is globally Lipschitz continuous and $\Pi _{\mathcal{S}_{+}^{|\beta |}}(Q \Lambda (Y)_{\beta \beta }Q^{T})=Q( \Lambda (Y)_{\beta \beta })_{+}Q^{T}$, we obtain that

$$\begin{aligned}&Q(\Lambda (Y)_{\beta \beta })_{+}Q^{T}-\Pi _{\mathcal{S}_{+}^{|\beta |}}(H_{\beta \beta })\\&\quad =Q(\Lambda (Y)_{\beta \beta })_{+}Q^{T}-\Pi _{\mathcal{S}_{+}^{|\beta |}}(Q \Lambda (Y)_{\beta \beta }Q^{T})+O(\Vert H\Vert ^{2})\\&\quad =O(\Vert H\Vert ^{2}). \end{aligned}$$

Therefore,

$$\begin{aligned} R(A+H)-R(A)-R'(A;H)=O(\Vert H\Vert ^{2}). \end{aligned}$$

(63)

By combining (59) and (63), we know that for any $\mathcal{S}^{n}\ni H\rightarrow 0$,

$$\begin{aligned} \Pi _{\mathcal{S}_{+}^{n}}(\Lambda (A)+H)-\Pi _{\mathcal{S}_{+}^{n}}(\Lambda (A))-\Pi '_{\mathcal{S}_{+}^{n}}(\Lambda (A);H)=O(\Vert H\Vert ^{2}). \end{aligned}$$

(64)

Next, consider the case that $A=\overline{P}^{T}\Lambda (A)\overline{P}$. Re-write (57) as

$$\begin{aligned} \Lambda (A)+\overline{P}^{T}H\overline{P}=\overline{P}^{T}P \Lambda (Y)P^{T}\overline{P}. \end{aligned}$$

Let $\widetilde{H}:=\overline{P}^{T}H\overline{P}$. Then, we have

$$\begin{aligned} \Pi _{\mathcal{S}^{n}_{+}}(A+H)=\overline{P}\,\Pi _{\mathcal{S}^{n}_{+}}(\Lambda (A)+\widetilde{H})\overline{P}^{T}. \end{aligned}$$

Therefore, since $\overline{P}\in \mathcal{O}^{n}$, we know from (64) and (14) that for any $\mathcal{S}^{n}\ni H\rightarrow 0$, (16) holds. $\square$

Proof of Proposition 3.3

Denote the set in the righthand side of (27) by $\mathcal N$. We first show that $N_{\mathrm{gph}\,N_{\mathcal{S}^{n}_+}} (0,0)\subseteq \mathcal{N}$. By the definition of the limiting normal cone in (8), we know that $(U^*,V^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^{|\beta |}_+}} (0,0)$ if and only if there exist two sequences $\{ ({U^k}^*,{V^k}^*)\}$ converging to $(U^*,V^*)$ and $\{(U^k,V^k)\}$ converging to $(0,0)$ with $({U^k}^*,{V^k}^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^{n}_+}}^\pi (U^k,V^k)$ and $(U^k,V^k)\in \mathrm{gph}\,N_{\mathcal{S}^{n}_+}$ for each $k$.

For each $k$, denote $A^{k}\!:=\!U^{k}\!+\!V^{k}\in \mathcal{S}^{n}$ and let $A^{k}\!=\!P^{k}\Lambda (A^{k})(P^{k})^{T}$ with $P^{k}\!\in \!\mathcal{O}^{n}$ be the eigenvalue decomposition of $A^{k}$. Then for any $i\!\in \!\{1,\ldots ,n\}$, we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \lambda _{i}(A^{k})=0. \end{aligned}$$

Since $\{P^k\}_{k=1}^{\infty }$ is uniformly bounded, by taking a subsequence if necessary, we may assume that $\{P^k\}_{k=1}^{\infty }$ converges to an orthogonal matrix $Q := \displaystyle {\lim \nolimits _{k\rightarrow \infty }} P^k\in \mathcal{O}^{n}$. For each $k$, we know that the vector $\lambda (A^{k})$ is an element of ${\mathfrak {R}}^{n}_{\gtrsim }$. By taking a subsequence if necessary, we may assume that for each $k$, $\Lambda (A^{k})$ has the same form, i.e.,

$$\begin{aligned} \Lambda (A^{k})=\left[ \begin{array}{l@{\quad }l@{\quad }l}\Lambda (A^{k})_{\beta _{+}\beta _{+}} &{} 0 &{} 0 \\ 0 &{} \Lambda (A^{k})_{\beta _{0}\beta _{0}} &{} 0 \\ 0 &{} 0 &{} \Lambda (A^{k})_{\beta _{-}\beta _{-}}\end{array}\right] , \end{aligned}$$

where $\beta _{+}$, $\beta _{0}$ and $\beta _{-}$ are the three index sets defined by

$$\begin{aligned} \beta _+:=\{i:\, \lambda _{i}(A^{k})>0\}, \quad \beta _0:=\{i:\, \lambda _{i}(A^{k})=0\}\quad \mathrm{and} \quad \beta _-:= \{i:\, \lambda _{i}(A^{k})<0\}. \end{aligned}$$

Since $({U^k}^*,{V^k}^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^{n}_+}}^\pi (U^k,V^k)$, we know from Proposition 3.2 that for each $k$, there exist

$$\begin{aligned} \Theta ^{k}_{1} =\left[ \begin{array}{l@{\quad }l@{\quad }l} E_ {\beta _{+} \beta _{+}} &{} E_{\beta _{+} \beta _{0}} &{} \Sigma _{\beta _{+} \beta _{-}}^{k}\\ E_{\beta _{+} \beta _{0}}^T &{} 0 &{} 0 \\ ({\Sigma }_{\beta _{+} \beta _{-}}^{k})^T &{} 0 &{} 0 \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} \Theta ^{k}_{2} =\left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} E_{\beta _{+}\beta _{-}}-\Sigma _{\beta _{+} \beta _{-}}^{k}\\ 0 &{} 0 &{} E_{\beta _{0}\beta _{-}} \\ (E_{\beta _{+}\beta _{-}}-\Sigma _{\beta _{+} \beta _{-}}^{k}) ^T &{} (E_{\beta _{0}\beta _{-}})^{T} &{} E_{\beta _{-}\beta _{-}} \end{array} \right] \end{aligned}$$

such that

$$\begin{aligned} \Theta _{1}^{k}\circ {\widetilde{U^{*}}}^k+\Theta _{2}^{k}\circ \widetilde{{V^{k}}}^*=0, \quad \widetilde{{U^{k}}}^*_{\beta _{0}\beta _{0}}\preceq 0 \quad \mathrm{and}\quad \widetilde{{V^{k}}}^*_{\beta _{0}\beta _{0}}\succeq 0, \end{aligned}$$

(65)

where $\widetilde{{U^{k}}}^*=(P^{k})^{T}{U^{k}}^*P^{k}$, $\widetilde{{V^{k}}}^*=(P^{k})^{T}{V^{k}}^*P^{k}$ and

$$\begin{aligned} (\Sigma ^{k})_{i,j}= \frac{\max \{ \lambda _i(A^{k}) ,0\}-\max \{ \lambda _j(A^{k}) ,0\}}{\lambda _i(A^{k})-\lambda _j(A^{k})} \quad \forall \,(i,j) \in \beta _{+}\times \beta _{-}. \end{aligned}$$

(66)

Since for each $k$, each element of ${\Sigma }^{k}_{\beta _{+}\beta _{-}}$ belongs to the interval $[0,1]$, by further taking a subsequence if necessary, we may assume that the limit of $\{ {\Sigma }^{k}_{\beta _{+}\beta _{-}} \}_{k=1}^{\infty }$ exists. Therefore, by the definition of $\mathcal{U}_{n}$ in (24), we know that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Theta _{1}^{k}= \Xi _{1}\in \mathcal{U}_{n} \quad \mathrm{and} \quad \lim _{k\rightarrow \infty }\Theta _{2}^{k}=\Xi _{2}, \end{aligned}$$

where $\Xi _{1}$ and $\Xi _{2}$ are given by (26). Therefore, we obtain from (65) that $(U^{*},V^{*})\in \mathcal N.$

The other direction, i.e., $N_{\mathrm{gph}\,N_{\mathcal{S}^{n}_+}} (0,0)\supseteq \mathcal{N}$ can be proved in a similar but simpler way to that of the second part of Theorem 3.1. We omit it here. $\square$

Proof of Theorem 3.1

“$\Longrightarrow$” Suppose that $(X^*,Y^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^n_+}} (X,Y)$. By the definition of the limiting normal cone in (8), we know that $(X^*,Y^*)=\displaystyle {\lim \nolimits _{k \rightarrow \infty }} ({X^k}^*,{Y^k}^*)$ with

$$\begin{aligned} ({X^k}^*,{Y^k}^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^n_+}}^\pi (X^k,Y^k)\quad k=1,2,\ldots , \end{aligned}$$

where $(X^k,Y^k) \rightarrow (X,Y)$ and $(X^k,Y^k)\in \mathrm{gph}\,N_{\mathcal{S}^n_+}$. For each $k$, denote $A^{k}:=X^{k}+Y^{k}$ and let $A^{k}=P^{k}\Lambda (A^{k})(P^{k})^{T}$ be the eigenvalue decomposition of $A^{k}$. Since $\Lambda (A) = \displaystyle {\lim \nolimits _{k\rightarrow \infty }} \Lambda (A^{k})$, we know that $\Lambda (A^{k})_{\alpha \alpha }\succ 0$, $\Lambda (A^{k})_{\gamma \gamma }\prec 0$ for k sufficiently large and $\displaystyle {\lim \nolimits _{k \rightarrow \infty }} \displaystyle {\Lambda (A^{k})_{\beta \beta }} = 0$.

Since $\{P^k\}_{k=1}^{\infty }$ is uniformly bounded, by taking a subsequence if necessary, we may assume that $\{P^k\}_{k=1}^{\infty }$ converges to an orthogonal matrix $\widehat{P}\in \mathcal{O}^{n}(A)$. We can write $\widehat{P}=\left[ \overline{P}_{\alpha } \ \ \overline{P}_{\beta }Q \ \ \overline{P}_{\gamma }\right]$, where $Q\in \mathcal {O}^{|\beta |}$ can be any $|\beta |\times |\beta |$ orthogonal matrix. By further taking a subsequence if necessary, we may also assume that there exists a partition $\pi (\beta )=(\beta _{+}, \beta _{0}, \beta _{-})$ of $\beta$ such that for each $k$,

$$\begin{aligned} \lambda _{i}(A^k) >0\quad \forall \, i\in \beta _{+}, \quad \lambda _{i}(A^k) =0\quad \forall \, i\in \beta _{0} \quad \text{ and } \quad \lambda _{i}(A^k) <0\quad \forall \, i\in \beta _{-}. \end{aligned}$$

This implies that for each $k$,

$$\begin{aligned} \{i:\lambda _{i}(A^{k})>0\}\!=\!\alpha \cup \beta _{+}, \{i:\lambda _{i}(A^{k})=0\}\!=\!\beta _{0} \quad \text{ and } \quad \{i:\lambda _{i}(A^{k})<0\}\!=\!\beta _{-}\cup \gamma . \end{aligned}$$

Then, for each $k$, since $({X^k}^*,{Y^k}^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^n_+}}^\pi (X^k,Y^k)$, we know from Proposition 3.2 that there exist

$$\begin{aligned} \Theta ^{k}_{1}=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} E_ {\alpha \alpha } &{} E_{\alpha \beta _{+}} &{} E_{\alpha \beta _{0}} &{} \Sigma _{\alpha \beta _{-}}^{k} &{} \Sigma _{\alpha \gamma }^{k}\\ E_{\alpha \beta _{+}}^T &{} E_{\beta _{+}\beta _{+}} &{} E_{\beta _{+}\beta _{0}} &{} \Sigma _{\beta _{+}\beta _{-}}^{k} &{} \Sigma _{\beta _{+}\gamma }^{k} \\ E_{\alpha \beta _{0}}^T &{} E_{\beta _{+}\beta _{0}}^{T} &{} 0 &{} 0 &{} 0 \\ {\Sigma ^{k}_{\alpha \beta _{-}}}^{T} &{} {\Sigma ^{k}_{\beta _{+}\beta _{-}}}^{T} &{} 0 &{} 0 &{} 0 \\ {\Sigma ^{k}_{\alpha \gamma }}^{T} &{} {\Sigma ^{k}_{\beta _{+}\gamma }}^{T} &{} 0 &{} 0 &{} 0 \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} \Theta ^{k}_{2}=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0 &{} E_{\alpha \beta _{-}}-\Sigma _{\alpha \beta _{-}}^{k} &{} E_{\alpha \gamma }-\Sigma _{\alpha \gamma }^{k}\\ 0 &{} 0 &{} 0 &{} E_{\beta _{+}\beta _{-}}-\Sigma _{\beta _{+}\beta _{-}}^{k} &{} E_{\beta _{+}\gamma }-\Sigma _{\beta _{+}\gamma }^{k} \\ 0 &{} 0 &{} 0 &{} E_{\beta _{0}\beta _{-}} &{} E_{\beta _{0}\gamma } \\ (E_{\alpha \beta _{-}}-\Sigma _{\alpha \beta _{-}}^{k})^{T} &{} (E_{\beta _{+}\beta _{-}}-\Sigma _{\beta _{+}\beta _{-}}^{k})^{T} &{} E_{\beta _{0}\beta _{-}}^{T} &{} E_{\beta _{-}\beta _{-}} &{} E_{\beta _{-}\gamma } \\ (E_{\alpha \gamma }-\Sigma _{\alpha \gamma }^{k})^{T} &{} (E_{\beta _{+}\gamma }-\Sigma _{\beta _{+}\gamma }^{k})^{T} &{} E_{\beta _{0}\gamma }^{T} &{} E_{\beta _{-}\gamma }^{T} &{} E_{\gamma \gamma } \end{array} \right] \end{aligned}$$

such that

$$\begin{aligned} \Theta _{1}^{k}\circ \widetilde{{X^{k}}}^*+\Theta _{2}^{k}\circ \widetilde{{Y^{k}}}^*=0, \quad \widetilde{{X^{k}}}^*_{\beta _{0}\beta _{0}}\preceq 0 \quad \mathrm{and} \quad \widetilde{{Y^{k}}}^*_{\beta _{0}\beta _{0}}\succeq 0, \end{aligned}$$

(67)

where $\widetilde{{X^{k}}}^*=(P^{k})^{T}{X^{k}}^*P^{k}, \widetilde{{Y^{k}}}^*=(P^{k})^{T}{Y^{k}}^*P^{k}$ and

$$\begin{aligned} (\Sigma ^{k})_{i,j}= \frac{\max \{ \lambda _i(A^{k}) ,0\}-\max \{ \lambda _j(A^{k}) ,0\}}{\lambda _i(A^{k})-\lambda _j(A^{k})} \quad \forall \, (i,j) \in (\alpha \cup \beta _{+}) \times (\beta _{-}\cup \gamma ). \end{aligned}$$

(68)

By taking limits as $k\rightarrow \infty$, we obtain that

$$\begin{aligned} \widetilde{{X^{k}}}^*\rightarrow \widehat{P}^{T}X^{*} \widehat{P}=\left[ \begin{array}{l@{\quad }l@{\quad }l} \widetilde{X}_{\alpha \alpha }^{*} &{} \widetilde{X}_{\alpha \beta }^{*}Q &{} \widetilde{X}_{\alpha \gamma }^{*}\\ (\widetilde{X}_{\alpha \beta }^{*}Q)^{T} &{} Q^{T}\widetilde{X}_{\beta \beta }^{*}Q &{} Q^{T}\widetilde{X}_{\beta \gamma }^{*}\\ (\widetilde{X}_{\alpha \gamma }^{*})^{T} &{} (Q^{T}\widetilde{X}_{\beta \gamma }^{*})^{T} &{} \widetilde{X}_{\gamma \gamma } \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} \widetilde{{Y^{k}}}^*\rightarrow \widehat{P}^{T}Y^{*} \widehat{P}=\left[ \begin{array}{l@{\quad }l@{\quad }l} \widetilde{Y}_{\alpha \alpha }^{*} &{} \widetilde{Y}_{\alpha \beta }^{*}Q &{} \widetilde{Y}_{\alpha \gamma }^{*}\\ (\widetilde{Y}_{\alpha \beta }^{*}Q)^{T} &{} Q^{T}\widetilde{Y}_{\beta \beta }^{*}Q &{} Q^{T}\widetilde{Y}_{\beta \gamma }^{*}\\ (\widetilde{Y}_{\alpha \gamma }^{*})^{T} &{} (Q^{T}\widetilde{Y}_{\beta \gamma }^{*})^{T} &{} \widetilde{Y}_{\gamma \gamma } \end{array} \right] . \end{aligned}$$

By simple calculations, we obtain from (68) that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Sigma _{\alpha \beta _{-}}^{k}=E_{\alpha \beta _{-}}, \quad \lim _{k\rightarrow \infty }\Sigma _{\beta _{+}\gamma }^{k}=0 \quad \text{ and } \quad \lim _{k\rightarrow \infty }\Sigma _{\alpha \gamma }^{k}=\Sigma _{\alpha \gamma }. \end{aligned}$$

This, together with the definition of $\mathcal{U}_{|\beta |}$, shows that there exist $\Xi _{1}\in \mathcal{U}_{|\beta |}$ and the corresponding $\Xi _{2}$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Theta _{1}^{k}=\left[ \begin{array}{l@{\quad }l@{\quad }l} E_{\alpha \alpha } &{} E_{\alpha \beta } &{} \Sigma _{\alpha \gamma }\\ E_{\beta \alpha } &{} \Xi _{1} &{} 0\\ \Sigma _{\alpha \gamma }^{T} &{} 0 &{} 0 \end{array} \right] =\Theta _{1}+\left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0 \\ 0 &{} \Xi _{1} &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} \lim _{k\rightarrow \infty }\Theta _{2}^{k}=\left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} E_{\alpha \gamma }-\Sigma _{\alpha \gamma }\\ 0 &{} \Xi _{2} &{} E_{\beta \gamma }\\ (E_{\alpha \gamma }-\Sigma _{\alpha \gamma })^{T} &{} E_{\gamma \beta } &{} E_{\gamma \gamma } \end{array} \right] =\Theta _{2}+\left[ \begin{array}{l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0 \\ 0 &{} \Xi _{2} &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right] , \end{aligned}$$

where $\Theta _{1}$ and $\Theta _{2}$ are given by (22). Meanwhile, since $Q\in \mathcal{O}^{|\beta |}$, by taking limits in (67) as $k\rightarrow \infty$, we obtain that

$$\begin{aligned} \Theta _{1}\circ \widetilde{X}^{*}+\Theta _{2}\circ \widetilde{Y}^{*}=0, \quad \Xi _{1}\circ Q^{T}\widetilde{X}_{\beta \beta }^{*}Q+\Xi _{2}\circ Q^{T}\widetilde{Y}_{\beta \beta }^{*}Q=0 \end{aligned}$$

(69)

and

$$\begin{aligned} Q_{\beta _{0}}^{T}{\widetilde{X}}_{\beta \beta }^{*}{Q_{\beta _{0}}}\preceq 0 \quad \text{ and } \quad Q_{\beta _{0}}^{T}{\widetilde{Y}}_{\beta \beta }^{*}{Q_{\beta _{0}}} \succeq 0. \end{aligned}$$

Hence, by Proposition 3.3, we conclude that $(\widetilde{X}_{\beta \beta }^{*}, \widetilde{Y}_{\beta \beta }^{*})\in N_{\mathrm{gph}\,N_{\mathcal{S}^{|\beta |}_+}} (0,0)$. From (69), it is easy to check that $(X^{*},Y^{*})$ satisfies the conditions (28) and (29).

“$\Longleftarrow$” Let $(X^{*},Y^{*})$ satisfies (28) and (29). We shall show that there exist two sequences $\{(X^k,Y^k)\}$ converging to $(X,Y)$ and $\{({X^k}^*,{Y^k}^*)\}$ converging to $(X^*,Y^*)$ with $(X^k,Y^k)\in \mathrm{gph}\,N_{\mathcal{S}^n_+}$ and $({X^k}^*,{Y^k}^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^n_+}}^\pi (X^k,Y^k)$ for each $k$.

Since $(\widetilde{X}_{\beta \beta }^{*}, \widetilde{Y}_{\beta \beta }^{*})\in N_{\mathrm{gph}\,N_{\mathcal{S}^{|\beta |}_+}} (0,0)$, by Proposition 3.3, we know that there exist an orthogonal matrix $Q\in \mathcal{O}^{|\beta |}$ and $\Xi _1\in \mathcal{U}_{|\beta |}$ such that

$$\begin{aligned} \Xi _{1}\circ Q^{T}\widetilde{X}_{\beta \beta }^{*}Q+\Xi _{2}\circ Q^{T}\widetilde{Y}_{\beta \beta }^{*}Q=0, \quad Q_{\beta _{0}}^{T}\widetilde{X}_{\beta \beta }^{*}Q_{\beta _{0}}\preceq 0 \quad \text{ and } \quad Q_{\beta _{0}}^{T}\widetilde{Y}_{\beta \beta }^{*} Q_{\beta _{0}}\succeq 0. \end{aligned}$$

(70)

Since $\Xi _1\in \mathcal{U}_{|\beta |}$, we know that there exists a sequence $\{z^{k}\}\in {\mathfrak {R}}^{|\beta |}_{\gtrsim }$ converging to $0$ such that $\Xi _1=\displaystyle {\lim \nolimits _{k\rightarrow \infty }}D(z^k)$. Without loss of generality, we can assume that there exists a partition $\pi (\beta )=(\beta _{+},\beta _{0},\beta _{-})\in \fancyscript{P}(\beta )$ such that for all $k$,

$$\begin{aligned} z_{i}^{k}>0 \quad \forall \, i\in \beta _{+}, \quad z_{i}^{k}=0 \quad \forall \, i\in \beta _{0} \quad \mathrm{and} \quad \quad z_{i}^{k}<0 \quad \forall \, i\in \beta _{-}. \end{aligned}$$

For each $k$, let

$$\begin{aligned} X^{k}=\widehat{P} \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \Lambda (A)_{\alpha \alpha } &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} (z^{k})_{+} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \end{array} \right] \widehat{P}^{T} \quad \mathrm{and} \quad Y^{k}\!=\!\widehat{P} \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} (z^{k})_{-} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \Lambda (A)_{\gamma \gamma } \end{array} \right] \widehat{P}^{T}\!, \end{aligned}$$

where $\widehat{P}=\left[ \overline{P}_{\alpha } \, \, \overline{P}_{\beta }Q \, \, \overline{P}_{\gamma }\right] \in \mathcal{O}^{n}(A)$. Then, it is clear that $\{(X^k,Y^k)\}\in \mathrm{gph}\,N_{\mathcal{S}^n_+}$ converging to $(X,Y)$. For each $k$, denote

$$\begin{aligned} A^{k}=X^{k}+Y^{k}, \quad \Theta ^{k}_{1}=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} E_ {\alpha \alpha } &{} E_{\alpha \beta _{+}} &{} E_{\alpha \beta _{0}} &{} \Sigma _{\alpha \beta _{-}}^{k} &{} \Sigma _{\alpha \gamma }\\ E_{\alpha \beta _{+}}^T &{} E_{\beta _{+}\beta _{+}} &{} E_{\beta _{+}\beta _{0}} &{} \Sigma _{\beta _{+}\beta _{-}}^{k} &{} \Sigma _{\beta _{+}\gamma }^{k} \\ E_{\alpha \beta _{0}}^T &{} E_{\beta _{+}\beta _{0}}^{T} &{} 0 &{} 0 &{} 0 \\ (\Sigma ^{k}_{\alpha \beta _{-}})^{T} &{} (\Sigma ^{k}_{\beta _{+}\beta _{-}})^{T} &{} 0 &{} 0 &{} 0 \\ (\Sigma _{\alpha \gamma })^{T} &{} ({\Sigma ^{k}}_{\beta _{+}\gamma })^{T} &{} 0 &{} 0 &{} 0 \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} \Theta ^{k}_{2}=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0 &{} 0 &{} 0 &{} E_{\alpha \beta _{-}}-\Sigma _{\alpha \beta _{-}}^{k} &{} E_{\alpha \gamma }-\Sigma _{\alpha \gamma }\\ 0 &{} 0 &{} 0 &{} E_{\beta _{+}\beta _{-}}-\Sigma _{\beta _{+}\beta _{-}}^{k} &{} E_{\beta _{+}\gamma }-\Sigma _{\beta _{+}\gamma }^{k} \\ 0 &{} 0 &{} 0 &{} E_{\beta _{0}\beta _{-}} &{} E_{\beta _{0}\gamma } \\ (E_{\alpha \beta _{-}}-\Sigma _{\alpha \beta _{-}}^{k})^{T} &{} (E_{\beta _{+}\beta _{-}}-\Sigma _{\beta _{+}\beta _{-}}^{k})^{T} &{} E_{\beta _{0}\beta _{-}}^{T} &{} E_{\beta _{-}\beta _{-}} &{} E_{\beta _{-}\gamma } \\ (E_{\alpha \gamma }-\Sigma _{\alpha \gamma })^{T} &{} (E_{\beta _{+}\gamma }-\Sigma _{\beta _{+}\gamma }^{k})^{T} &{} E_{\beta _{0}\gamma }^{T} &{} E_{\beta _{-}\gamma }^{T} &{} E_{\gamma \gamma } \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} (\Sigma ^{k})_{i,j}= \frac{\max \{ \lambda _i(A^{k})) ,0\}-\max \{ \lambda _j(A^{k})) ,0\}}{\lambda _i(A^{k})-\lambda _j(A^{k})} \quad \forall \, (i,j) \in (\alpha \cup \beta _{+}) \times (\beta _{-}\cup \gamma ). \end{aligned}$$

Next, for each $k$, we define two matrices $\widehat{{X^{k}}}^*, \widehat{{Y^{k}}}^*\in \mathcal{S}^{n}$. Let $i,j\in \{1,\ldots ,n\}$. If $(i,j)$ and $(j,i)\notin (\alpha \times \beta _{-}) \cup (\beta _{+}\times \gamma )\cup (\beta \times \beta )$. We define

$$\begin{aligned} {\widehat{{X^{k}}}^*}_{i,j}\equiv { \widetilde{X}^*}_{i,j}, \quad { \widehat{{Y^{k}}}^*}_{i,j}\equiv { \widetilde{Y}^*}_{i,j},\quad k=1,2,\ldots . \end{aligned}$$

(71)

Otherwise, denote $c^{k}:=(\Sigma ^{k})_{i,j}$, $k=1,2,\ldots$. We consider the following four cases.

Case 1 $(i,j)$ or $(j,i)\in \alpha \times \beta _{-}$. In this case, we know from (28) that ${\widetilde{X}^*}_{i,j}=0$. Since $c_{k}\ne 0$ for all $k$ and $c^{k}\rightarrow 1$ as $k\rightarrow \infty$, we define

$$\begin{aligned} {\widehat{{Y^{k}}}^*}_{i,j}\equiv {\widetilde{Y}^*}_{i,j}\quad \mathrm{and} \quad {\widehat{{X^{k}}}^*}_{i,j}=\frac{c^{k}-1}{c^{k}}{\widehat{{Y^{k}}}^*}_{i,j},\quad k=1,2,\ldots . \end{aligned}$$

(72)

Then, we have

$$\begin{aligned} c_{k}{\widehat{{X^{k}}}^*}_{i,j}+(1-c_{k}){\widehat{{Y^{k}}}^*}_{i,j}=0 \quad \forall \, k \quad \mathrm{and} \quad ({\widehat{{X^{k}}}^*}_{i,j},{\widehat{{Y^{k}}}^*}_{i,j})\rightarrow ({\widetilde{X}^*}_{i,j},{\widetilde{Y}^*}_{i,j})\quad \mathrm{as}\ k\rightarrow \infty . \end{aligned}$$

Case 2 $(i,j)$ or $(j,i)\in \beta _{+}\times \gamma$. In this case, we know from (28) that ${\widetilde{Y}^*}_{i,j}=0$. Since $c_{k}\ne 1$ for all $k$ and $c^{k}\rightarrow 0$ as $k\rightarrow \infty$, we define

$$\begin{aligned} {\widehat{{X^{k}}}^*}_{i,j}\equiv {\widetilde{X}^*}_{i,j}\quad \mathrm{and} \quad {\widehat{{Y^{k}}}^*}_{i,j}=\frac{c^{k}}{c^{k}-1}{\widehat{{X^{k}}}^*}_{i,j}, \quad k=1,2,\ldots . \end{aligned}$$

(73)

Then, we know that

$$\begin{aligned} c_{k}{\widehat{{X^{k}}}^*}_{i,j}+(1-c_{k}){\widehat{{Y^{k}}}^*}_{i,j}=0 \quad \forall \, k \quad \mathrm{and} \quad ({\widehat{{X^{k}}}^*}_{i,j},{\widehat{{Y^{k}}}^*}_{i,j})\rightarrow ({\widetilde{X}^*}_{i,j},{\widetilde{Y}^*}_{i,j})\quad \mathrm{as}\ k\rightarrow \infty . \end{aligned}$$

Case 3 $(i,j)$ or $(j,i)\in (\beta \times \beta ){\setminus } (\beta _{+}\times \beta _{-})$. In this case, we define

$$\begin{aligned} {\widehat{{X^{k}}}^*}_{i,j} \equiv Q_{i}^{T}{\widetilde{X}^*}_{\beta \beta }Q_{j}, \quad {\widehat{{Y^{k}}}^*}_{i,j}\equiv Q_{i}^{T}{\widetilde{Y}^*}_{\beta \beta }Q_{j},\quad k=1,2,\ldots . \end{aligned}$$

(74)

Case 4 $(i,j)$ or $(j,i)\in \beta _{+}\times \beta _{-}$. Since $c\in [0,1]$, we consider the following two sub-cases:

Case 4.1 $c\ne 1$. Since $c_{k}\ne 1$ for all $k$ large enough, we define

$$\begin{aligned} {\widehat{{X^{k}}}^*}_{i,j}\equiv Q_{i}^{T}{\widetilde{X}^*}_{\beta \beta }Q_{j}\quad \mathrm{and} \quad {\widehat{{Y^{k}}}^*}_{i,j}=\frac{c^{k}}{c^{k}-1}{\widehat{{X^{k}}}^*}_{i,j}, \quad k=1,2,\ldots . \end{aligned}$$

(75)

Then, from (70), we know that

$$\begin{aligned} {\widehat{{Y^{k}}}^*}_{i,j}\rightarrow \frac{c}{c-1}Q_{i}^{T}{\widetilde{X}^*}_{\beta \beta }Q_{j}=Q_{i}^{T}{\widetilde{Y}^*}_{\beta \beta }Q_{j}\quad \mathrm{as}\ k\rightarrow \infty . \end{aligned}$$

Case 4.2 $c=1$. Since $c_{k}\ne 0$ for all $k$ large enough, we define

$$\begin{aligned} {\widehat{{Y^{k}}}^*}_{i,j}\equiv Q_{i}^{T}{\widetilde{Y}^*}_{\beta \beta }Q_{j}\quad \mathrm{and} \quad {\widehat{{X^{k}}}^*}_{i,j}=\frac{c^{k}-1}{c^{k}}{\widehat{{Y^{k}}}^*}_{i,j}, \quad k=1,2,\ldots . \end{aligned}$$

(76)

Then, again from (70), we know that

$$\begin{aligned} {\widehat{{X^{k}}}^*}_{i,j}\rightarrow \frac{c-1}{c}Q_{i}^{T}{\widetilde{Y}^*}_{\beta \beta }Q_{j}=Q_{i}^{T}{\widetilde{X}^*}_{\beta \beta }Q_{j}\quad \mathrm{as}\ k\rightarrow \infty . \end{aligned}$$

For each $k$, define ${X^k}^*=\widehat{P} \widehat{{X^{k}}}^*\widehat{P}^{T}$ and ${Y^k}^*=\widehat{P} \widehat{{Y^{k}}}^*\widehat{P}^{T}$. Then, from (71)–(76) we obtain that

$$\begin{aligned} \Theta _{1}^{k}\circ \widehat{P}^{T}{X^{k}}^{*}\widehat{P}+\Theta _{2}^{k}\circ \widehat{P}^{T}{Y^{k}}^{*}\widehat{P}=0,\quad k=1,2,\ldots . \end{aligned}$$

and

$$\begin{aligned} (\widehat{P}^{T}{X^{k}}^{*}\widehat{P}, \widehat{P}^{T}{Y^{k}}^{*}\widehat{P})\rightarrow (\widehat{P}^{T}{X}^{*}\widehat{P}, \widehat{P}^{T}{Y}^{*}\widehat{P})\quad \mathrm{as}\ k\rightarrow \infty . \end{aligned}$$

(77)

Moreover, from (74) and (70), we have

$$\begin{aligned}&Q_{\beta _{0}}^{T}{\widetilde{{X^{k}}}^*}_{\beta \beta }Q_{\beta _{0}}\equiv Q_{\beta _{0}}^{T}\widetilde{X}_{\beta \beta }^{*}Q_{\beta _{0}}\preceq 0 \quad \mathrm{and} \quad Q_{\beta _{0}}^{T}{\widetilde{{Y^{k}}}^*}_{\beta \beta }Q_{\beta _{0}} \equiv Q_{\beta _{0}}^{T}\widetilde{Y}_{\beta \beta }^{*} Q_{\beta _{0}}\succeq 0,\\&\quad k=1,2,\ldots . \end{aligned}$$

From Proposition 3.2 and (77), we know that

$$\begin{aligned} ({X^k}^*,{Y^k}^*)\in N_{\mathrm{gph}\,N_{\mathcal{S}^n_+}}^\pi (X^k,Y^k)\quad \mathrm{and} \quad (X^*,Y^*)=\displaystyle {\lim \limits _{k \rightarrow \infty }} ({X^k}^*,{Y^k}^*). \end{aligned}$$

Hence, the assertion of the theorem follows.