Optimality Conditions and Stability Analysis for the Second-Order Cone Constrained Variational Inequalities

Abstract

In this paper, we study the optimality conditions and perform a stability analysis for the second-order cone constrained variational inequalities (SOCCVI) problem. The Lagrange function and Karush–Kuhn–Tucker (KKT) condition of the SOCCVI problem is given, and the optimality conditions for the SOCCVI problem are studied. Then, the second-order sufficient condition satisfying the constrained nondegenerate condition is proved. The strong second-order sufficient condition is defined. And the nonsingularity of Clarke’s generalized Jacobian of the KKT point, the strong regularity of the KKT point, the uniform second-order growth condition, the strong stability of the KKT point, and the local Lipschtiz homeomorphism of the KKT point for the SOCCVI problem are proved to be equivalent to each other. Then, the stability theorem of the SOCCVI problem is obtained.

1. Introduction

In this paper, we consider the second-order cone constrained variational inequalities (SOCCVI) problem: finding $x^{*}\in \mathrm{\Omega }$ such that

$$\langle F\left(x^{*}\right),y-x^{*}\rangle \ge 0,-g\left(y\right)\in \mathcal{K},y\in \mathrm{\Omega },$$

(1)

where $\langle ·,·\rangle$ is the Euclidean inner product, $F:{\Re }^n\to {\Re }^n$, $g:{\Re }^n\to {\Re }^m$ are continuously differentiable, $\mathrm{\Omega }\subset {\Re }^n$ is a closed convex set, and $\mathcal{K}$ is the Cartesian product of second-order cones (SOCs) as follows:

$$\mathcal{K}={\mathcal{K}}^{m_1}\times {\mathcal{K}}^{m_2}\times \cdots \times {\mathcal{K}}^{m_p},$$

(2)

where $m_i\ge 1$, ${\sum }_{i=1}^p{m}_i=m$, and ${\mathcal{K}}^{m_i}:=\left\{s=(s_0;\overline{s})\in \Re \times {\Re }^{m_i-1}:s_0\ge \parallel \overline{s}\parallel \right\}$ is a second-order cone in ${\Re }^{m_i}$. We denote $g\left(y\right)=(g^{m_1}\left(y\right),\cdots ,g^{m_p}\left(y\right))$ and $g^{m_i}\left(y\right)=(g_0^{m_i}\left(y\right);{\overline{g}}^{m_i}\left(y\right)):{\Re }^n\to {\Re }^{m_i}$ for $i\in \left\{1,\cdots ,p\right\}$. Then, we have the following equivalent relations:

$$-g\left(y\right)\in \mathcal{K}\iff -g^{m_i}\left(y\right)\in {\mathcal{K}}^{m_i},\forall i\in \{1,\cdots ,p\}\iff g_0^{m_i}\left(x\right)\le -\parallel {\overline{g}}^{m_i}\left(x\right)\parallel ,\forall i\in \{1,\cdots ,p\}.$$

Variational inequalities are widely used in physics, mechanics, economics, optimization, control, and equilibrium models in transportation, etc. The second-order cone constrained variational inequalities problem has also attracted the attention of many scholars. Recently, Sun et al. [1] constructed an implementable augmented Lagrangian method for solving the second-order cone constrained variational inequalities (SOCCVI) problem (1). Nazemi and Sabeghi [2] reduced the variational inequalities to a convex second-order cone programming (CSOCP) problem and solved the convex second-order cone constrained variational inequalities problem by using a gradient neural network model. Nazemi and Sabeghi [3] used a new neural network model as a simple solution to the convex second-order cone constrained variational inequalities problem.

The optimality condition is a necessary and sufficient condition that the objective function and the constraint function must satisfy at the optimal point of the optimization problem. The optimality conditions play an important role in the termination rule of the algorithm and the proof of optimization theory. In order to ensure the convergence of algorithms, we should analyze the stability of the solution to the optimization problems (or perturbation analysis). Stability analysis was used for the first time in the study of the sensitivity analysis of linear programming by Manne [4]. Danskin [5] computed the directional differentiability and directional derivative at the optimal solution. Bonnans and Ramirez [6] proved that the strong regularity of the KKT point for the second-order cone programming (SOCP) problem is equivalent to the constrained nondegenerate condition and the strong second-order sufficient condition, which was the perturbation analysis of the second-order cone programming problems. Wang and Zhang [7] studied the stability analysis for the second-order cone programming (SOCP) problem and proved that the nonsingularity of Clarke generalized Jacobian and the strong regularity of the KKT point for the non-smooth KKT system are equivalent to each other.

In this paper, we shall study the KKT condition, the first-order necessary condition, the second-order sufficient condition, and Clarke’s generalized differential nonsingularity of the non-smooth mapping corresponding to the KKT condition for the second-order cone constrained variational inequalities (SOCCVI) problem (1). We will demonstrate the relationship between the strong second-order sufficient condition and the nonsingularity of Clarke’s generalized Jacobian for the KKT system of the SOCCVI problem. And the strong regularity of the KKT point, the strong stability of the KKT point, and the local Lipschitz homeomorphism near the KKT point will be equivalent to each other.

The content of this paper is outlined as follows: In Section 2, we give some preliminaries that will be used in the following sections. Section 3 studies the KKT conditions of the SOCCVI problem (1) and proves the first-order necessary condition, the second-order sufficient condition, and the Clarke generalized differential nonsingularity of the non-smooth mapping corresponding to the KKT conditions. Finally, we establish and prove the relationship between the strong second-order sufficient condition, the nonsingularity of Clarke’s generalized differentiation, the strong regularity of the KKT point, the strong stability of the KKT point, and the local Lipschitz homeomorphism near the KKT point for the SOCCVI problems in Section 4.

2. Preliminaries

The projection operator to a convex set is quite useful in reformulating the box-constrained variational inequality as a non-smooth equation. Let C be a convex closed set, and for every $x\in {\Re }^n$, there is a unique $\widehat{x}$ in C such that

$$\parallel x-\widehat{x}\parallel =min\{\parallel x-y\parallel |y\in C\}.$$

The point $\widehat{x}$ is the projection of x onto C, denoted by ${\mathrm{\Pi }}_C\left(x\right)$. The projection operator ${\mathrm{\Pi }}_C:{\Re }^n\to C$ is well defined over ${\Re }^n$, and it is a nonexpensive mapping.

Lemma 1

([8]). Let H be a real Hilbert space and $C\subset H$ be a closed convex set. For a given $z\in H,$$u\in C$ satisfies the inequality

$$\langle u-z,v-u\rangle \ge 0,\forall v\in C$$

if and only if $u-{\mathrm{\Pi }}_C\left(z\right)=0$.

More information on the projection operator can be found in Section 3 of the book by Goebel and Reich [9].

The second-order cone (SOC) in ${\Re }^n$ ($n\ge 1$), also called the Lorentz cone or the ice-cream cone, is defined as

$${\mathcal{K}}^n=\{(x_0;\overline{x})|x_0\in \Re ,\overline{x}\in {\Re }^{n-1},x_0\ge \parallel \overline{x}\parallel \}.$$

If $n=1$, ${\mathcal{K}}^n$ is the set of nonnegative reals ${\Re }_{+}$.

For any two vectors for $x=(x_0;\overline{x})\in \Re \times {\Re }^{n-1}$ and $y=(y_0;\overline{y})\in \Re \times {\Re }^{n-1}$, the Jordan product of x and y is denoted by

$$x\circ y=(x^{\mathrm{T}}y;x_0\overline{y}+y_0\overline{x}).$$

As a result, the normal addition “+”, “∘”, and $e=(1;0)$ produce the Jordan algebra $(\Re \times {\Re }^{n-1},\circ )$ associated with the second-order cone.

Any $x=(x_0;\overline{x})\in \Re \times {\Re }^{n-1}$ has the following spectral decomposition (see Faraut and Kor$\acute{\mathrm{a}}$nyi [10]).

$$x={\lambda }_1\left(x\right)c_1\left(x\right)+{\lambda }_2\left(x\right)c_2\left(x\right),$$

(3)

where ${\lambda }_1\left(x\right)$ and ${\lambda }_2\left(x\right)$ are the spectral values of x with formulas

$${\lambda }_i\left(x\right)=x_0+{(-1)}^i\parallel \overline{x}\parallel ,(i=1,2),$$

(4)

while $c_1\left(x\right)$ and $c_2\left(x\right)$ are the spectral vectors associated with x given by

$$c_i\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\left(1;{(-1)}^i{\displaystyle \frac{\overline{x}}{\parallel \overline{x}\parallel }}\right),\hfill & \overline{x}\ne 0,\hfill \\ \hfill \\ {\displaystyle \frac{1}{2}}(1;{(-1)}^i\omega ),\hfill & \overline{x}=0,\hfill \end{array}\right.$$

(5)

where $\omega \in {\Re }^{n-1}$, $\parallel \omega \parallel =1$, and $i=1,2$.

The projection of x onto ${\mathcal{K}}^n$, denoted by ${\mathrm{\Pi }}_{{\mathcal{K}}^n}$, can be represented as follows:

$${\mathrm{\Pi }}_{{\mathcal{K}}^n}={\left[{\lambda }_1\left(x\right)\right]}_{+}{c}_1\left(x\right)+{\left[{\lambda }_2\left(x\right)\right]}_{+}{c}_2\left(x\right),$$

where ${\left[{\lambda }_i\left(x\right)\right]}_{+}=\max \{0,{\lambda }_i\left(x\right)\},i=1,2.$ We can calculate it as follows:

$${\mathrm{\Pi }}_{{\mathcal{K}}^n}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}(1+{\displaystyle \frac{x_0}{\parallel \overline{x}\parallel }})(\parallel \overline{x}\parallel ;\overline{x}),\hfill & |x_0|<\parallel \overline{x}\parallel ,\hfill \\ x,\hfill & \parallel \overline{x}\parallel \le x_0,\hfill \\ 0,\hfill & \parallel \overline{x}\parallel \le -x_0.\hfill \end{array}\right.$$

Importantly, ${\mathrm{\Pi }}_{{\mathcal{K}}^n}(·)$ is strongly semi-smooth over ${\Re }^n$ (see Chen et al. [11]).

In order to obtain the stability analysis for the SOCCVI problem (1), we recall the differential properties in the following.

Lemma 2

([12]). ${\mathrm{\Pi }}_{{\mathcal{K}}^n}(·)$ is directionally differentiable at x for any $h\in {\Re }^n$. Moreover, the directional derivative is described by

$${{\mathrm{\Pi }}^{\prime }}_{{\mathcal{K}}^n}(x;h)=\left\{\begin{array}{cc}\mathcal{J}{\mathrm{\Pi }}_{{\mathcal{K}}^n}\left(x\right)h,\hfill & x\in {\Re }^n\setminus ({\mathcal{K}}^n\cup -{\mathcal{K}}^n),\hfill \\ h-2{\left[c_1{\left(x\right)}^{\mathrm{T}}h\right]}_{-}{c}_1\left(x\right),\hfill & x\in \mathrm{bd}{\mathcal{K}}^n\setminus \left\{0\right\},\hfill \\ 2{\left[c_2{\left(x\right)}^{\mathrm{T}}h\right]}_{+}{c}_2\left(x\right),\hfill & x\in -\mathrm{bd}{\mathcal{K}}^n\setminus \left\{0\right\},\hfill \\ {\mathrm{\Pi }}_{{\mathcal{K}}^n}\left(h\right),\hfill & x=0,\hfill \end{array}\right.$$

where

$$\begin{array}{c}\mathcal{J}{\mathrm{\Pi }}_{{\mathcal{K}}^n}\left(x\right)={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& {\displaystyle \frac{{\overline{x}}^{\mathrm{T}}}{\parallel \overline{x}\parallel }}\\ {\displaystyle \frac{\overline{x}}{\parallel \overline{x}\parallel }}& I_{n-1}+{\displaystyle \frac{x_0}{\parallel \overline{x}\parallel }}{I}_{n-1}-{\displaystyle \frac{x_0}{\parallel \overline{x}\parallel }}{\displaystyle \frac{\overline{x}{\overline{x}}^{\mathrm{T}}}{\parallel \overline{x}{\parallel }^2}}\end{array}\right),\hfill \end{array}$$

${\left[c_1{\left(x\right)}^{\mathrm{T}}h\right]}_{-}=\min \{0,\left[c_1{\left(x\right)}^{\mathrm{T}}h\right]\}$, and ${\left[c_2{\left(x\right)}^{\mathrm{T}}h\right]}_{+}=\max \{0,\left[c_2{\left(x\right)}^{\mathrm{T}}h\right]\}.$

For the convenience of further discussion, we need the definition of the tangent cone, regular normal cone, and the second-order tangent cone of a closed set at a point. All the concepts are found in Rockafellar and Wets [13].

Definition 1

([13]). For a closed set $K\subset {\mathcal{R}}^n$ and a point $x\in K$, the following sets are defined.

The tangent (Bouligand) cone is as follows:

$$T_K\left(x\right):={\displaystyle \underset{t\searrow 0}{limsup}}{\displaystyle \frac{K-x}{t}},$$

where $t\searrow 0$ represents that t decreases towards 0. The regular normal cone is as follows:

$${\widehat{N}}_K\left(x\right):=\{v\in {\Re }^n|\langle v,y-x\rangle \le \mathtt{o}(\parallel y-x\parallel ),\forall y\in K\}.$$

The limiting (in the sense of Mordukhovich) normal cone is as follows:

$$N_K\left(x\right):={limsup}_{\stackrel{K}{y\to x}}{\widehat{N}}_K\left(y\right).$$

If K is a closed convex set, then $T_K\left(x\right)=\mathtt{cl}(K+\Re x)$ and ${\widehat{N}}_K\left(x\right)=N_K\left(x\right)=T_K{\left(x\right)}^{\circ }=\{v\in K^{\circ }|\langle v,x\rangle \le 0\},$ where $K^{\circ }$ denotes the polar of K. According to Bonnans and Shapiro [14] and the definition of the tangent, we obtain that for any $d\in T_K\left(x^{*}\right)$, there exist $t_k\searrow 0$ and $d^k\to d$ such that $x^{*}+t_k{d}^k\in K$, and for $k\to \infty ,z^{*}+t_k{d}^k\in K\cap {\mathbb{B}}_{\epsilon }\left(x^{*}\right)$.

Lemma 3

([13]). The definitions of the tangent cone and the second-order tangent cone of ${\mathcal{K}}^n$ at $x\in {\mathcal{K}}^n$ are

$$T_{{\mathcal{K}}^n}\left(x\right)=\left\{\begin{array}{cc}{\Re }^n,\hfill & x\in \mathrm{int}{\mathcal{K}}^n,\hfill \\ {\mathcal{K}}^n,\hfill & x=0,\hfill \\ \{d=(d_0,\overline{d})\in {\Re }^n\times {\Re }^{n-1}|\langle \overline{d},\overline{x}\rangle -x_0{d}_0\le 0\},\hfill & x\in \mathrm{bd}{\mathcal{K}}^n\setminus \left\{0\right\}.\hfill \end{array}\right.$$

and

$$T_{{\mathcal{K}}^n}^2(x,d)=\left\{\begin{array}{cc}{\Re }^n,\hfill & x\in \mathrm{int}{T}_{{\mathcal{K}}^n},\hfill \\ T_{{\mathcal{K}}^n}\left(d\right),\hfill & x=0,\hfill \\ \{w=(w_0,\overline{w})\in {\Re }^n\times {\Re }^{n-1}|\langle \overline{w},\overline{s}\rangle -w_0{x}_0\le d_0^2-\parallel \overline{d}{\parallel }^2\},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Next, we recall the smoothing metric projection operator as follows:

$$\mathrm{\Phi }(\epsilon ,x):={\displaystyle \frac{1}{2}}(x+\sqrt{{\epsilon }^{2}e+x^2})$$

(6)

for any $(\epsilon ,x)\in \Re \times {\Re }^n$, where $e=(1,0,\cdots ,0)\in {\Re }^n$ and $x^2=x\circ x$. Notice that when $\epsilon =0$, then $\mathrm{\Phi }(0,x)={\mathrm{\Pi }}_{{\mathcal{K}}^n}\left(x\right)$, and $\mathrm{\Phi }(\epsilon ,x)$ is continuously differentiable at any $(\epsilon ,x)\in \Re \times {\Re }^n$ if ${({\epsilon }^{2}e+x^2)}_0\ne \parallel \overline{{\epsilon }^{2}e+x^2}\parallel$. Moreover, $\mathrm{\Phi }(·,·)$ is globally Lipschitz continuous and strongly semi-smooth for any $(0,x)\in \Re \times {\Re }^n$.

Let x have the spectral decomposition $u={\lambda }_1\left(x\right)c_1\left(x\right)+{\lambda }_2\left(x\right)c_2\left(x\right)$. For simplicity, we denote ${\lambda }_i={\lambda }_i\left(x\right),c_i=c_i\left(x\right)$ for $i=1,2$. It is easy to verify that $\mathrm{\Phi }$ has the following spectral decomposition:

$$\begin{array}{c}\mathrm{\Phi }(\epsilon ,x)=\varphi (\epsilon ,{\lambda }_1)c_1+\varphi (\epsilon ,{\lambda }_2)c_2\hfill \end{array}$$

(7)

where $\varphi (\epsilon ,·):={\displaystyle \frac{1}{2}}(·+\sqrt{{\epsilon }^2+{(·)}^2})$, for any $(\epsilon ,·)\in \Re \times \Re$. Thus, the function $\mathrm{\Phi }$ has the following expression:

$$\mathrm{\Phi }(\epsilon ,x)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}x+{\displaystyle \frac{1}{4}}\left(\begin{array}{c}\sqrt{{\epsilon }^2+{\lambda }_2^2}+\sqrt{{\epsilon }^2+{\lambda }_1^2}\\ \left(\sqrt{{\epsilon }^2+{\lambda }_2^2}-\sqrt{{\epsilon }^2+{\lambda }_1^2}\right){\displaystyle \frac{\overline{x}}{\parallel \overline{x}\parallel }}\end{array}\right),\hfill & if\overline{x}\ne 0,\hfill \\ \hfill \\ {\displaystyle \frac{1}{2}}\left(\begin{array}{c}{x}_0+\sqrt{{\epsilon }^2+x_0^2}\\ 0\end{array}\right),\hfill & if\overline{x}=0.\hfill \end{array}\right.$$

(8)

If ${({\epsilon }^{2}e+x^2)}_0\ne \parallel \overline{{\epsilon }^{2}e+x^2}\parallel$, we can calculate the derivative of $\mathrm{\Phi }(\epsilon ,x)$ as follows:

$$\mathcal{J}\mathrm{\Phi }(\epsilon ,x)=\left({\mathcal{J}}_{\epsilon }\mathrm{\Phi }(\epsilon ,x){\mathcal{J}}_x\mathrm{\Phi }(\epsilon ,x)\right),$$

where

$$\begin{array}{c}{\mathcal{J}}_{\epsilon }\mathrm{\Phi }(\epsilon ,x)={\displaystyle \frac{1}{2}}\left({\mathcal{J}}_{\epsilon }\varphi (\epsilon ,{\lambda }_1)c_1+{\mathcal{J}}_{\epsilon }\varphi (\epsilon ,{\lambda }_2)c_2\right)\hfill \\ ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\epsilon c_1}{\sqrt{{\epsilon }^2+{\lambda }_1^2}}}+{\displaystyle \frac{\epsilon c_2}{\sqrt{{\epsilon }^2+{\lambda }_2^2}}}\right),\hfill \end{array}$$

(9)

and

(i) if $\overline{x}\ne 0$, we have

$$\begin{array}{c}{\mathcal{J}}_x\mathrm{\Phi }(\epsilon ,x)={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\lambda }_1}{\sqrt{{\epsilon }^2+{\lambda }_1^2}}}+{\displaystyle \frac{{\lambda }_2}{\sqrt{{\epsilon }^2+{\lambda }_2^2}}}\right)& Y^{\mathrm{T}}\\ Y& Z\end{array}\right),\hfill \end{array}$$

(10)

where

$$Y={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\lambda }_2}{\sqrt{{\epsilon }^2+{\lambda }_2^2}}}-{\displaystyle \frac{{\lambda }_1}{\sqrt{{\epsilon }^2+{\lambda }_1^2}}}\right){\displaystyle \frac{\overline{x}}{\parallel \overline{x}\parallel }}$$

and

$$\begin{array}{c}Z=\left[1+{\displaystyle \frac{\sqrt{{\epsilon }^2+{\lambda }_2^2}-\sqrt{{\epsilon }^2+{\lambda }_1^2}}{{\lambda }_2-{\lambda }_1}}\right]I_{n-1}\\ +\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\lambda }_1}{\sqrt{{\epsilon }^2+{\lambda }_1^2}}}+{\displaystyle \frac{{\lambda }_2}{\sqrt{{\epsilon }^2+{\lambda }_2^2}}}\right)-{\displaystyle \frac{\sqrt{{\epsilon }^2+{\lambda }_2^2}-\sqrt{{\epsilon }^2+{\lambda }_1^2}}{{\lambda }_2-{\lambda }_1}}\right]{\displaystyle \frac{{\overline{x}}^{\mathrm{T}}\overline{x}}{\parallel \overline{x}{\parallel }^2}}\end{array}$$

(ii) if $\overline{x}=0$, we obtain that

$$\begin{array}{c}{\mathcal{J}}_x\mathrm{\Phi }(\epsilon ,x)=\left\{{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{x_0}{\sqrt{{\epsilon }^2+x_0^2}}}\right)I_n\right\}.\hfill \end{array}$$

(11)

If ${({\epsilon }^{2}e+x^2)}_0=\parallel \overline{{\epsilon }^{2}e+x^2}\parallel$, we can calculate the B-subdifferential of $\mathrm{\Phi }$ as follows:

Lemma 4

([15]). If ${({\epsilon }^{2}e+x^2)}_0=\parallel \overline{{\epsilon }^{2}e+x^2}\parallel$ and ${\epsilon }^2+x_0^2+{\parallel \overline{x}\parallel }^2=2|x_0|\parallel \overline{x}\parallel$, ${\partial }_B\mathrm{\Phi }(\epsilon ,x)$ can be calculated as follows:

(i) if $\epsilon =0,x_0=\parallel \overline{x}\parallel \ne 0$, we have

$$\begin{array}{c}{\partial }_B\mathrm{\Phi }(\epsilon ,x)=\left\{\left(\pm \sqrt{\gamma (1-\gamma )}{c}_1{I}_n-2\gamma c_1{c}_1^{\mathrm{T}}\right):\gamma \in [0,1]\right\}.\hfill \end{array}$$

(12)

(ii) if $\epsilon =0,-x_0=\parallel \overline{x}\parallel \ne 0$, we obtain that

$$\begin{array}{c}{\partial }_B\mathrm{\Phi }(\epsilon ,x)=\left\{\left(\pm \sqrt{\gamma (1-\gamma )}{c}_{2}2\gamma c_2{c}_2^{\mathrm{T}}\right):\gamma \in [0,1]\right\}.\hfill \end{array}$$

(13)

(iii) if $\epsilon =x_0=\parallel \overline{x}\parallel =0$, we obtain that

$$\begin{array}{c}{\partial }_B\mathrm{\Phi }(\epsilon ,x)=\left\{\left(\pm \sqrt{\gamma (1-\gamma )e}\gamma e{e}^{\mathrm{T}}\right):\gamma \in [0,1]\right\}\hfill \\ \cup \left\{V({\gamma }_1,{\gamma }_2,\alpha ,\omega )|\begin{array}{c}\exists {\epsilon }_k\downarrow 0,\exists {\overline{x}}^k\to 0,{\overline{x}}^k\ne 0,{\displaystyle \frac{{\overline{x}}^k}{\parallel {\overline{x}}^k\parallel }}\to \omega ,\\ \alpha ({\epsilon }_k,x^k)\to \alpha ,{\gamma }_i({\epsilon }_k,x^k)\to {\gamma }_i,i=1,2\end{array}\right\},\hfill \end{array}$$

(14)

where

$$\alpha ({\epsilon }_k,x^k):={\displaystyle \frac{\varphi ({\epsilon }_k,{\lambda }_2^k)-\varphi ({\epsilon }_k,{\lambda }_1^k)}{{\lambda }_2^k-{\lambda }_1^k}},{\gamma }_i({\epsilon }_k,u^k):={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{{\lambda }_i^k}{\sqrt{{\epsilon }_k^2+{\left({\lambda }_i^k\right)}^2}}}\right),$$

$$V({\gamma }_1,{\gamma }_2,\alpha ,\omega ):=\left(v({\gamma }_1,{\gamma }_2,\alpha ,\omega )\mathrm{\Gamma }({\gamma }_1,{\gamma }_2,\alpha ,\omega )\right),$$

(15)

where $v({\gamma }_1,{\gamma }_2,\alpha ,\omega )\in {\Re }^n$ and $\mathrm{\Gamma }({\gamma }_1,{\gamma }_2,\alpha ,\omega )\in {\Re }^{n\times n}$ are defined by

$$v({\gamma }_1,{\gamma }_2,\alpha ,\omega )=\pm {\displaystyle \frac{1}{2}}\left(\sqrt{{\gamma }_1(1-{\gamma }_1)}\left(\begin{array}{c}1\hfill \\ -\omega \hfill \end{array}\right)+\sqrt{{\gamma }_2(1-{\gamma }_2)}\left(\begin{array}{c}1\hfill \\ \omega \hfill \end{array}\right)\right),$$

(16)

$$\mathrm{\Gamma }({\gamma }_1,{\gamma }_2,\alpha ,\omega )={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}{\gamma }_1+{\gamma }_2\hfill & ({\gamma }_2-{\gamma }_1){\omega }^{\mathrm{T}}\hfill \\ ({\gamma }_2-{\gamma }_1)\omega \hfill & 2\alpha (I_{n-1}-\omega {\omega }^{\mathrm{T}})+({\gamma }_1+{\gamma }_2)\omega {\omega }^{\mathrm{T}}\hfill \end{array}\right).$$

(17)

Now, we give an expression of $\alpha$ in terms of ${\gamma }_1$ and ${\gamma }_2$. Let $\beta (\epsilon ,x):=\sqrt{{\displaystyle \frac{{\epsilon }^2+{\lambda }_1^2}{{\epsilon }^2+{\lambda }_2^2}}}$. If $\exists {\epsilon }_k\downarrow 0$, $\exists {\overline{x}}^k\to 0$, ${\overline{x}}^k\ne 0$, ${\displaystyle \frac{{\overline{x}}^k}{\parallel {\overline{x}}^k\parallel }}\to \omega$, $\alpha ({\epsilon }_k,x^k)\to \alpha$, ${\gamma }_i({\epsilon }_k,x^k)\to {\gamma }_i$, for $i=1,2$, we can obtain that $\beta ({\epsilon }_k,x^k)\to \beta$ for some $\beta \in [0,+\infty ]$. Therefore, we have the following:

$$\alpha =\left\{\begin{array}{cc}{\displaystyle \frac{{\gamma }_2+\beta {\gamma }_1}{1+\beta }},\hfill & \beta \in [0,+\infty ),\hfill \\ \hfill \\ {\gamma }_1,\hfill & \beta =+\infty \hfill \end{array}\right.$$

(18)

The following conclusions will play an important role in the subsequent discussions on optimality and stability.

Lemma 5

([16]). Suppose that $x\in \mathcal{Q}$ and $z\in \mathcal{Q}$ (i.e., $x_i{⪰}_{\mathcal{Q}}0$ and $z_i{⪰}_{\mathcal{Q}}0$, for all i = 1,⋯,r). Then, $x^{\mathrm{T}}z$=0, if and only if $x_i\circ z_i$ = Arw($x_i$)Arw($z_i$)e=0 for i = 1, ⋯, r, or equivalently,

(i) 

$x_i^{\mathrm{T}}{z}_i=x_{i0}{z}_{i0}+{\overline{x}}_i^{\mathrm{T}}{\overline{z}}_i=0$, $i=1,\cdots ,r$

(ii) 

$x_{i0}{\overline{z}}_i+z_{i0}{\overline{x}}_i=0$, $i=1,\cdots ,r$

Lemma 6

([17]). For any $V_0\in {\partial }_B{\mathrm{\Pi }}_Q\left(u\right)$, there exists $V\in {\partial }_B\mathrm{\Phi }(0,u)$ with $V=\left(v\mathrm{\Gamma }\right)$ such that

$$V_0=\mathrm{\Gamma }.$$

Proposition 1

([17]). Let $(x^{*},{\mu }^{*})$ be a Karush–Kuhn–Tucker point. Then, for any $V\in \partial \mathrm{\Phi }(0,-g\left(x^{*}\right)-{\mu }^{*})$ and $\Delta u,\Delta v\in {\Re }^m$ such that $\Delta u=V(0,\Delta u+\Delta v)$, it holds that

$$\langle \Delta u,\Delta v\rangle \ge S(x^{*},\mu ),$$

where $S(x^{*},\mu )=S^{m_1}(x^{*},\mu )\times S^{m_2}(x^{*},\mu )\times \cdots \times S^{m_i}(x^{*},\mu )$, and $S^{m_i}(x^{*},\mu )={\displaystyle \sum _{j=1}^{mi}}{S}^j(x^{*},\mu )$ is defined by

$$S^{m_i}(x^{*},\mu ):=\left\{\begin{array}{cc}-{\displaystyle \frac{{\left({\mu }_{m_i}\right)}_0}{(-g_0^{m_i}\left(x^{*}\right))}}{\left({(\Delta u)}^{m_i}\right)}^{\mathrm{T}}\left(\begin{array}{cc}1\hfill & 0^{\mathrm{T}}\hfill \\ 0\hfill & -I_{m_i}\hfill \end{array}\right){(\Delta u)}^{m_i},\hfill & m_i\in B^{*}\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right\},$$

(19)

where $B^{*}$ is a case of the index set, and the specific index set is as follows:

$$\begin{array}{c}{I}^{*}:=\left\{m_i\right|-g^{m_i}\left(x^{*}\right)\in \mathrm{int}{\mathcal{K}}^{mi}\},\hfill \\ Z^{*}:=\left\{m_i\right|g^{m_i}\left(x^{*}\right)=0\},\hfill \\ B^{*}:=\left\{m_i\right|-g^{m_i}\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^{mi}\setminus \left\{0\right\}\},\hfill \end{array}$$

(20)

where $i=1,\cdots ,p$.

3. Optimality Condition

The Lagrange function of the SOCCVI problem (1) is as follows:

$$\mathcal{L}(x,\mu )=F\left(x\right)+\mathcal{J}g{\left(x\right)}^{\mathrm{T}}\mu ,$$

(21)

where $\mu =({\mu }^{m_1},{\mu }^{m_2},\cdots ,{\mu }^{m_p})\in \mathcal{K},{\mu }^{m_i}\in {\mathcal{K}}^{m_i}(i=1,2,\cdots ,p)$.

The KKT condition of the SOCCVI problem (1) is as follows:

$$\left\{\begin{array}{c}\mathcal{L}(x,\mu )=F\left(x\right)+\mathcal{J}g{\left(x\right)}^{\mathrm{T}}\mu =0,\hfill \\ -g\left(x\right)\in \mathcal{K},\mu \in \mathcal{K},\hfill \\ -g\left(x\right)\circ \mu =0,\hfill \end{array}\right.$$

(22)

where “∘” is the Jordan product.

Let us consider the second-order cone constrained optimization (SOCCOP) problem.

$$\begin{array}{c}minf\left(x\right)\hfill \\ \mathrm{s}.\mathrm{t}.-g\left(x\right)\in \mathcal{K},\hfill \end{array}$$

(23)

where $\mathcal{K}={\mathcal{K}}^{m_1}\times {\mathcal{K}}^{m_2}\times \cdots \times {\mathcal{K}}^{m_p}$ is defined by (2), $f:{\Re }^n\to \Re$ is a twice continuously differentiable function, and $g:{\Re }^n\to {\Re }^m$ is a twice continuously differentiable mapping.

It is easy to know that the Lagrange function of the SOCCOP problem (21) can be written as follows:

$${\mathcal{L}}_{min}(x,\mu )=f\left(x\right)+\langle \mu ,g\left(x\right)\rangle ,$$

where $\mu =({\mu }^{m_1},{\mu }^{m_2},\cdots ,{\mu }^{m_p})\in \mathcal{K},{\mu }^{m_i}\in {\mathcal{K}}^{m_i}(i=1,2,\cdots ,p)$.

The KKT condition of the SOCCOP problem (23) is as follows:

$$\left\{\begin{array}{c}{\mathcal{J}}_x{\mathcal{L}}_{min}(x,\mu )=\nabla f\left(x\right)+\mathcal{J}g{\left(x\right)}^{\mathrm{T}}\mu =0,\hfill \\ -g\left(x\right)\in \mathcal{K},\mu \in \mathcal{K},\hfill \\ -g\left(x\right)\circ \mu =0.\hfill \end{array}\right.$$

(24)

Clearly, if $\nabla f\left(x\right)=F\left(x\right)$, the KKT mappings of the SOCCOP problem (23) and the SOCCVI problem (1) are the same. Moreover, if f is a twice continuously differentiable convex function, these two problems are equivalent. Due to the close relationship between the SOCCVI problem (1) and SOCCOP (23), we will study the optimality conditions and stability analysis of the SOCCVI problem (1) in the framework of SOCCOP (23). The following research is assumed to be true of the condition $\nabla f\left(x\right)=F\left(x\right)$.

If $(x^{*},{\mu }^{*})$ satisfies the KKT condition (21), then $x^{*}$ is a stationary point of the SOCCVI problem (1), and the set of multipliers is denoted by $\mathrm{\Lambda }\left(x^{*}\right)$. If the constraint nondegeneracy condition holds at $x^{*}$, that is,

$$-\mathcal{J}g\left(x^{*}\right){\Re }^n+\mathrm{lin}{T}_{\mathcal{K}}(-g\left(x^{*}\right))={\Re }^m,$$

then $x^{*}$ is nondegenerate.

Suppose that $x^{*}$ is a stationary point of the SOCCVI problem (1); then, Robinson’s constraint qualification holds at $x^{*}$; that is,

$$0\in \mathrm{int}\{-g\left(x^{*}\right)-\mathcal{J}g\left(x^{*}\right){\Re }^n-\mathcal{K}\},$$

(25)

and the Lagrange multiplier set $\mathrm{\Lambda }\left(x^{*}\right)$ is nonempty compact.

It follows from Corollary 26 in Bonnans and Ramirez [6] that we can deduce the critical cone at a stable point $x^{*}$ of the SOCCVI problem (1).

$$\mathcal{C}\left(x^{*}\right)=\left\{d\right|-\mathcal{J}g\left(x^{*}\right)d\in T_{\mathcal{K}}(-g\left(x^{*}\right)),\langle F\left(x^{*}\right),d\rangle =0\}.$$

(26)

According to Definition 1 and Lemma 5, the above critical cone $\mathcal{C}\left(x^{*}\right)$ can be expressed as follows:

$$\mathcal{C}\left(x^{*}\right)=\left\{d\in {\Re }^n|\begin{array}{cc}-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d\in T_{{\mathcal{K}}^{m_i}}(-g^{m_i}\left(x^{*}\right)),\hfill & {\mu }^{m_i}=0\hfill \\ -\mathcal{J}{g}^{m_i}\left(x^{*}\right)d=0,\hfill & {\mu }^{m_i}\in \mathrm{int}{\mathcal{K}}^{m_i}\hfill \\ -\mathcal{J}{g}^{m_i}\left(x^{*}\right)d\in \in \Re ({\mu }_0^{m_i};-{\overline{\mu }}^{m_i}),\hfill & {\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\},-g^{m_i}\left(x^{*}\right)=0\hfill \\ \langle -\mathcal{J}{g}^{m_i}\left(x^{*}\right),d\rangle =0,\hfill & {\mu }^{m_i},-g^{m_i}\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}\hfill \end{array}\right\}.$$

Next, we will obtain the second-order sufficient condition for the stationary point of the SOCCVI problem (1).

Theorem 1.

Suppose that $x^{*}$ is a stationary point of the SOCCVI problem (1) and satisfies the Robinson’s Constraint Qualification (25); then, the second-order growth condition holds at $x^{*}$ if and only if the second-order sufficient condition holds at $x^{*}$; that is,

$${\displaystyle \underset{\mu \in \mathrm{\Lambda }\left(x^{*}\right)}{sup}}\{\langle {\mathcal{J}}_x\mathcal{L}(x^{*},\mu )d,d\rangle +d^{\mathrm{T}}\mathcal{H}(x^{*},\mu )d\}>0,\forall d\in \mathcal{C}\left(x^{*}\right)\setminus \left\{0\right\},$$

(27)

where $\mathcal{H}(x^{*},\mu )={\mathcal{H}}^{m_1}(x^{*},\mu )\times {\mathcal{H}}^{m_2}(x^{*},\mu )\times \cdots \times {\mathcal{H}}^{m_p}(x^{*},\mu )$, ${\mathcal{H}}^{m_i}(x^{*},{\mu }^i)={\displaystyle \sum _{j=1}^{m_i}}{\mathcal{H}}^j(x^{*},\mu ),(i=1,\cdots ,p)$, and

$${\mathcal{H}}^j(x^{*},\mu )=\left\{\begin{array}{cc}{\displaystyle \frac{{\mu }_0^j}{g_0^j\left(x^{*}\right)}}\mathcal{J}{g}^j{\left(x^{*}\right)}^{\mathrm{T}}\left(\begin{array}{cc}1\hfill & 0^{\mathrm{T}}\hfill \\ 0\hfill & -I_j\hfill \end{array}\right)\mathcal{J}{g}^j\left(x^{*}\right),\hfill & -g^j\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^j\setminus \left\{0\right\},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(28)

Proof. 

If the second-order growth condition holds, and the set $\mathcal{K}$ is the second-order regular at $-g\left(x^{*}\right)$ along $\mathcal{J}g\left(x^{*}\right)d$, then we obtain that

$${\displaystyle \underset{\mu \in \mathrm{\Lambda }\left(x^{*}\right)}{sup}}\{\langle {\mathcal{J}}_x\mathcal{L}(x^{*},\mu )d,d\rangle -\sigma (-\mu ;{\mathcal{T}}^2)\}>0,\forall d\in \mathcal{C}\left(x^{*}\right)\setminus \left\{0\right\},$$

(29)

where ${\mathcal{T}}^2:={\mathcal{T}}_{\mathcal{K}}^2(-g\left(x^{*}\right),-\mathcal{J}g\left(x^{*}\right)d)$ is the second-order tangent set at $-g\left(x^{*}\right)$ along the direction $-\mathcal{J}g\left(x^{*}\right)d$, and $\sigma (·;{\mathcal{T}}^2)$ is the support function of the set ${\mathcal{T}}^2$. Therefore, we only prove that (27) and (29) are equivalent.

According to the definition of the second-order tangent set, we set $h^{m_i}\left(d\right)=-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d$ and $s^{m_i}=-g^{m_i}\left(x^{*}\right)$). Then, we have the following from Lemma 4:

$$\begin{array}{c}{\mathcal{T}}_i^2=T_{{\mathcal{K}}_{mi}}^2(s^{m_i},h^{m_i}\left(d\right))\hfill \\ =\left\{\begin{array}{cc}{\Re }^{m_i},\hfill & h^{m_i}\left(d\right)\in \mathrm{int}{T}_{{\mathcal{K}}^{mi}}\left(s^{m_i}\right),\hfill \\ T_{{\mathcal{K}}^{m_i}}\left(h^{m_i}\left(d\right)\right),\hfill & s^{m_i}=0,\hfill \\ \{\omega \in {\Re }^{m_i}:{\overline{\omega }}^{\mathrm{T}}{\overline{s}}^{m_i}-{\omega }_0{s}_0^{m_i}\le h_0^{m_i}{\left(d\right)}^2-\parallel {\overline{h}}^{m_i}\left(d\right){\parallel }^2\},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(30)

Since $d\in \mathcal{C}\left(x^{*}\right)$, we obtain that $h^{m_i}\left(d\right)\in T_{{\mathcal{K}}^{mi}}\left(s^{m_i}\right)$. According to the definition of the second-order tangent set, $-{\mu }^{m_i}\in N_{{\mathcal{K}}^{m_i}}\left(s^{m_i}\right)$ and $s^{m_i}\in {\mathcal{K}}^{m_i}$, we obtain that

$$\sigma (-{\mu }^{m_i};{\mathcal{T}}_{m_i}^2)\le 0.$$

If $0\in {\mathcal{T}}_{m_i}^2$, we obtain that $\sigma (-{\mu }^{m_i};{\mathcal{T}}_{m_i}^2)=0$. From (30), when the condition $h^{m_i}\left(d\right)\in \mathrm{int}{T}_{{\mathcal{K}}^{mi}}\left(s^{m_i}\right)$, $s^{m_i}=0$, or $h^{m_i}\left(d\right)=0$ holds, we have $0\in {\mathcal{T}}_{m_i}^2$.

Next, considering $s^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$ and $h^{m_i}\left(d\right)\in \mathrm{bd}{T}_{{\mathcal{K}}^{mi}}\left(s^{m_i}\right)$, we let $\alpha :=\left\{m_i\right|s^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\},h^{m_i}\left(d\right)\in \mathrm{bd}{T}_{{\mathcal{K}}^{mi}}\left(s^{m_i}\right)\}$. From (30), for $m_i\in \alpha$, we can obtain that

$$\sigma (-{\mu }^{m_i};{\mathcal{T}}_{m_i}^2)={\displaystyle \underset{\omega \in {\Re }^{m_i}}{sup}}\{-({\omega }_0{\mu }_0^{m_i}+{\overline{\omega }}^{\mathrm{T}}{\overline{\mu }}^{m_i})|{\overline{\omega }}^{\mathrm{T}}\overline{s^{m_i}}-{\omega }_0{s}_0^{m_i}\le h_0^{m_i}{\left(d\right)}^2-\parallel {\overline{h}}^{m_i}\left(d\right){\parallel }^2\}.$$

(31)

From the KKT condition ${\mu }^{m_i}\circ s^{m_i}=0$, we obtain that ${\overline{\mu }}^{m_i}=-({\mu }_0^{m_i}/s_0^{m_i}){\overline{s}}^{m_i}$. Then, we have

$$-({\omega }_0{\mu }_0^{m_i}+{\overline{\omega }}^{\mathrm{T}}{\overline{\mu }}^{m_i})=({\mu }_0^{m_i}/s_0^{m_i})({\overline{\omega }}^{\mathrm{T}}{\overline{s}}^{m_i}-{\omega }_0{s}_0^{m_i}).$$

For $m_i\in \alpha$, we know that

$$\sigma (-{\mu }^{m_i};{\mathcal{T}}_{m_i}^2)={\displaystyle \frac{{\mu }_0^{m_i}}{s_0^{m_i}}}(h_0^{m_i}{\left(d\right)}^2-\parallel {\overline{h}}^{m_i}\left(d\right){\parallel }^2).$$

Since $\sigma (-\mu ;{\mathcal{T}}^2)={\displaystyle \sum _{i=1}^p}\sigma (-{\mu }^{m_i};{\mathcal{T}}_{m_i}^2)$, we have

$$\sigma (-\mu ;{\mathcal{T}}^2)={\displaystyle \sum _{m_i\in \alpha }}{\displaystyle \frac{{\mu }_0^{m_i}}{s_0^{m_i}}}(h_0^{m_i}{\left(d\right)}^2-\parallel {\overline{h}}^{m_i}\left(d\right){\parallel }^2).$$

Thus, we show that (27) and (29) are equivalent; that is, the second-order sufficient condition at $x^{*}$ is as follows:

$${\displaystyle \underset{\mu \in \mathrm{\Lambda }\left(x^{*}\right)}{sup}}\{\langle {\mathcal{J}}_x\mathcal{L}(x^{*},\mu )d,d\rangle +d^{\mathrm{T}}\mathcal{H}(x^{*},\mu )d\}>0,\forall d\in \mathcal{C}\left(x^{*}\right)\setminus \left\{0\right\}.$$

This completes the proof. □

If $x^{*}$ is a stationary point of the SOCCVI problem (1), the Robinson’s constraint qualification is satisfied at $x^{*}$; that is,

$$0\in \mathrm{int}\{-g\left(x^{*}\right)-\mathcal{J}g\left(x^{*}\right){\Re }^n-\mathcal{K}\}.$$

From the definition of normal cone, we have the Karush–Kuhn–Tucker condition.

$$\begin{array}{c}\mathcal{L}(x^{*},\mu )=F\left(x^{*}\right)+\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\mu =0,\hfill \\ -\mu \in N_{\mathcal{K}}(-g\left(x^{*}\right)),\hfill \end{array}$$

(32)

where $\mu \in {\Re }^m$.

The set of multipliers is denoted by $\mathrm{\Lambda }\left(x^{*}\right)$, which is a nonempty closed convex and bounded set. Then, the second expression can be as follows:

$${\mu }^{m_i}\perp (-g^{m_i}\left(x^{*}\right))\in {\mathcal{K}}^{m_i}.$$

The Constraint Nondegenerate Condition of the SOCCVI problem (1) holds at $x^{*}$ in the following:

$$-\mathcal{J}g\left(x^{*}\right){\Re }^n+\mathrm{lin}\left(T_{\mathcal{K}}(-g\left(x^{*}\right))\right)={\Re }^m.$$

(33)

In order to prove the stability theorem, the affine space of the critical cone and the strong second-order sufficient condition are defined below.

Let $x^{*}$ be a locally optimal solution of the SOCCVI problem (1); then, the affine space of the critical cone at $x^{*}$ is as follows:

$$\mathrm{aff} \mathcal{C}\left(x^{*}\right)=\left\{d\in {\Re }^n|\begin{array}{cc}-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d=0,\hfill & m_i\in S_1\hfill \\ \langle {\left({\mu }^{*}\right)}^{m_i},-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d\rangle =0,\hfill & m_i\in S_2\hfill \\ -\mathcal{J}{g}^{m_i}\left(x^{*}\right)d\in \Re ({\left({\mu }_{m_i}^{*}\right)}_0;-{\overline{\mu }}_{m_i}^{*}),\hfill & m_i\in N_2\hfill \end{array}\right\},$$

(34)

where the index set is defined by the following:

$$\begin{array}{c}{S}_1:=\left\{m_i\right|g^{m_i}\left(x^{*}\right)=0,{\mu }_{m_i}^{*}\in \mathrm{int}{\mathcal{K}}^{m_i}\},\hfill \\ S_2:=\left\{m_i\right|-g^{m_i}\left(x^{*}\right),{\mu }_{m_i}^{*}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}\},\hfill \\ S_3:=\left\{m_i\right|-g^{m_i}\left(x^{*}\right)\in \mathrm{int}{\mathcal{K}}^{m_i},{\mu }_{m_i}^{*}=0\},\hfill \\ N_1:=\left\{m_i\right|g^{m_i}\left(x^{*}\right)=0,{\mu }_{m_i}^{*}=0\},\hfill \\ N_2:=\left\{m_i\right|g^{m_i}\left(x^{*}\right)=0,{\mu }_{m_i}^{*}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}\},\hfill \\ N_3:=\left\{m_i\right|-g^{m_i}\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\},{\mu }_{m_i}^{*}=0\}.\hfill \end{array}$$

(35)

Definition 2.

Let $x^{*}$ be a feasible point of the SOCCVI problem (1) such that $\mathrm{\Lambda }\left(x^{*}\right)=\left\{({\lambda }^{*},{\mu }^{*})\right\}$ is a singleton. We say that the strong second-order sufficient condition holds at $x^{*}$ if

$$\langle d,{\mathcal{J}}_x\mathcal{L}(x^{*},{\mu }^{*})d\rangle +\langle d,\mathcal{H}(x^{*},{\mu }^{*})d\rangle >0,\forall d\in \mathrm{aff}\left(C\left(x^{*}\right)\right)\setminus \left\{0\right\},$$

(36)

where $\mathcal{H}(x^{*},{\mu }^{*}):={\mathcal{H}}^{m_1}(x^{*},{\mu }^{*})\times {\mathcal{H}}^{m_2}(x^{*},{\mu }^{*})\cdots \times {\mathcal{H}}^{m_p}(x^{*},{\mu }^{*})$, ${\mathcal{H}}^{m_i}(x^{*},{\mu }^{*})={\displaystyle \sum _{j=1}^{mi}}{\mathcal{H}}^j(x^{*},\mu ),(i=1,2,\cdots ,p)$ and

$${\mathcal{H}}^{m_i}(x^{*},\mu ):=\left\{\begin{array}{cc}-{\displaystyle \frac{{\left({\mu }_{m_i}\right)}_0}{g_0^{m_i}\left(x^{*}\right)}}\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}\left(\begin{array}{cc}1\hfill & 0^{\mathrm{T}}\hfill \\ 0\hfill & -I_{m_i}\hfill \end{array}\right)(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)),\hfill & m_i\in B^{*},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(37)

4. Stability Analysis

Let $x^{*}$ be a stationary point of the SOCCVI problem (1). Then, $\mathcal{L}(x^{*},{\mu }^{*})$ satisfies the Karush–Kuhn–Tucker condition (32) if and only if $Q(x^{*},{\mu }^{*})=0$, where

$$Q(x,\mu ):=\left(\begin{array}{c}F\left(x\right)+\mathcal{J}g{\left(x\right)}^{\mathrm{T}}\mu \hfill \\ -g\left(x\right)-{\mathrm{\Pi }}_{\mathcal{K}}(-g\left(x\right)-\mu )\hfill \end{array}\right).$$

(38)

The smoothing function $E:\Re \times {\Re }^n\times {\Re }^m\to \Re \times {\Re }^n\times {\Re }^m$ of the Karush–Kuhn–Tucker function $Q(x,\mu )$ is defined as follows:

$$E(\epsilon ,x,\mu )=\left(\begin{array}{c}\epsilon \hfill \\ F\left(x\right)+\mathcal{J}g{\left(x\right)}^{\mathrm{T}}\mu \hfill \\ -g\left(x\right)-\mathrm{\Phi }(\epsilon ,-g(x)-\mu )\hfill \end{array}\right),$$

(39)

where $\mathrm{\Phi }$ is defined in (6). It is obvious that

$$Q(x,\mu )=0⟺E(\epsilon ,x,\mu )=0.$$

Moreover, if $(x^{*},{\mu }^{*})$ is a Karush–Kuhn–Tucker point of the SOCCVI problem (1), then $E(0,x^{*},{\mu }^{*})=0$.

Next, we introduce the concepts and theorems related to stability analysis.

Let $(x^{*},{\mu }^{*})$ be the Karush–Kuhn–Tucker point of the SOCCVI problem (1); then, $Q(x^{*},{\mu }^{*})=0$ if and only if $(x^{*},{\mu }^{*})$ is a solution of the following inclusion:

$$0\in \left(\begin{array}{c}\mathcal{L}(x^{*},{\mu }^{*})\\ -g\left(x^{*}\right)\end{array}\right)+\left(\begin{array}{c}{N}_{{\Re }^n}\left(x^{*}\right)\\ N_{\mathcal{K}}\left({\mu }^{*}\right)\end{array}\right).$$

(40)

The Equation (40) can be described equivalently as the following generalized equation:

$$0\in \varphi \left(z\right)+N_{\mathcal{K}}\left(z\right),$$

(41)

where $z\in Z={\Re }^n\times {\Re }^m$, $z=(x,\mu )$, and $\varphi$ is a continuous differentiable map defined by

$$\varphi \left(z\right)=\left(\begin{array}{c}\mathcal{L}(x,\mu )\\ -g\left(x\right)\end{array}\right).$$

(42)

Next, we give the definition of the strongly regular solution of the KKT condition.

Definition 3.

We say that $z^{*}=(x^{*},{\mu }^{*})$ is the strongly regular solution of the KKT condition. If there exist a neighborhood V of $(x^{*},{\mu }^{*})$ and a neighborhood $B\in {\Re }^n\times {\Re }^m$ of $(0^n\times 0^m)$ such that for any $\delta :=({\delta }_1,{\delta }_2)\in B$, then the following linearization system

$$\delta \in \varphi \left(z^{*}\right)+\mathcal{J}\varphi \left(z^{*}\right)(z-z^{*})+N_D\left(z^{*}\right)$$

(43)

has a unique solution $z_V\left(\delta \right)=(x_V\left({\delta }_1\right),{\mu }_V\left({\delta }_2\right))\in V$, and this solution is Lipschtiz continuous.

The perturbation problem of the SOCCOP problem (23) is as follows:

$$\begin{array}{c}minf(x,u)\hfill \\ \mathrm{s}.\mathrm{t}.-g(x,u)\in \mathcal{K},\hfill \end{array}$$

(44)

where $\mathcal{U}$ is a finite-dimensional space, $u\in \mathcal{U}$, and $f(·,·):{\Re }^n\times \mathcal{U}\to \Re$, $g(·,·):{\Re }^n\times \mathcal{U}\to {\Re }^m$ are second continuously differentiable, satisfying $f(·,0)=f(·)$, $-g(·,0)=-g(·)$. We denote by ${\mathrm{SOCCOP}}_u$ the corresponding perturbation problem of the SOCCOP problem (23).

Similar to the discussion in Bonnans and Shapiro [14], we give the uniform second-order growth condition for the stationary point of the SOCCOP problem (23).

Definition 4.

Let $x^{*}$ be a stationary point of the SOCCOP problem (23). Suppose $\left(f\right(x,u),-g(x,u\left)\right)$ is a ${\mathcal{C}}^2$-smooth parametrization. The uniform second-order growth condition holds at $x^{*}$ if there exist $\alpha >0$, a neighborhood V of $x^{*}$, and a neighborhood $U\subset \mathcal{U}$ of $0^m$ such that, for any $u\in U$ and any stationary point $x\left(u\right)\in V$ of the ${SOCCOP}_u$ problem (44), the following inequality holds:

$$f(x,u)\ge {f(x\left(u\right),u)+\alpha \parallel x-x\left(u\right)\parallel }^2,\forall x\in V,-g(x,u)\in \mathcal{K}.$$

(45)

If (45) holds for any ${\mathcal{C}}^2$-smooth parametrization, then the uniform second-order growth condition holds at $x^{*}$.

Further, we can give the definition of a strong stable point as in Bonnans and Shapiro [14].

Definition 5.

Let $x^{*}$ be a stationary point of the SOCCOP problem (23). If, for any ${\mathcal{C}}^2$-smooth parametrization $\left(f\right(x,u),-g(x,u\left)\right)$, there exist the neighborhood V of $x^{*}$ and the neighborhood $U\subset \mathcal{U}$ of $0^m$ such that, for any $u\in \mathcal{U}$, the ${SOCCOP}_u$ problem (44) has a unique stationary point $x\left(u\right)\in V$, and $x(·)$ is continuously at $\mathcal{U}$, then $x^{*}$ is strongly stable.

The next theorem demonstrates the relationship between the strong second-order sufficient condition under the constraint nondegeneracy condition and the nonsingularity of the generalized Jacobian of the smoothing mapping E defined by (39).

Theorem 2.

Suppose that $(x^{*},{\mu }^{*})$ is a Karush–Kuhn–Tucker point of the SOCCVI problem (1), and the constraint nondegeneracy condition and the strong second-order sufficient condition (37) hold at $x^{*}$; then, any element in $\partial E(0,x^{*},{\mu }^{*})$ is nonsingular.

Proof. 

Let W be an arbitrary element in $\partial E(0,x^{*},{\mu }^{*})$, and $(\Delta \epsilon ,\Delta x,\Delta \mu )\in \Re \times {\Re }^n\times {\Re }^m$ satisfy

$$W\left(\begin{array}{c}\Delta \epsilon \hfill \\ \Delta x\hfill \\ \Delta \mu \hfill \end{array}\right)=0;$$

then, there exists $V\in \partial \mathrm{\Phi }(0,-g\left(x^{*}\right)-{\mu }^{*})$ satisfying

$$W\left(\begin{array}{c}\Delta \epsilon \hfill \\ \Delta x\hfill \\ \Delta \mu \hfill \end{array}\right)=\left(\begin{array}{c}\Delta \epsilon \hfill \\ {\mathcal{J}}_x\mathcal{L}(x^{*},{\mu }^{*})\Delta x+\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta \mu \hfill \\ \mathcal{J}g\left(x^{*}\right)\Delta x+V(\Delta \epsilon ,-\mathcal{J}g\left(x^{*}\right)\Delta x-\Delta \mu )\hfill \end{array}\right)=0.$$

(46)

Obviously, we have $\Delta \epsilon =0$. Thus, the third equation in (46) becomes

$$\mathcal{J}g\left(x^{*}\right)\Delta x+V(\Delta \epsilon ,-\mathcal{J}g\left(x^{*}\right)\Delta x-\Delta \mu )=0.$$

We will discuss the following three cases:

(a) If $m_i\in S_1$, we know that $g^{m_i}\left(x^{*}\right)=0$ and ${\mu }_{m_i}^{*}\in {\mathcal{K}}^{m_i}$. Then, we obtain $V^{m_i}\in \partial {\mathrm{\Phi }}^{m_i}(0,-g^{m_i}\left(x^{*}\right)-{\mu }_{m_i}^{*}=\left\{0\right\}$. Thus, for any $m_i\in S_1$, we obtain

$$-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x=0.$$

(b) If $m_i\in S_2$, we obtain $-g^{m_i}\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$ and ${\mu }_{m_i}^{*}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$. Then, there exists ${\xi }_{m_i}>0$ such that ${\mu }_{m_i}^{*}={\xi }^{m_i}(-g_0^{m_i}\left(x^{*}\right);{\overline{g}}^{m_i}\left(x^{*}\right))$, and we have

$$-g^{m_i}\left(x^{*}\right)-{\mu }_{m_i}^{*}=\left(-g_0^{m_i}\left(x^{*}\right)-{\left({\mu }_{m_i}^{*}\right)}_0;{\displaystyle \frac{-g_0^{m_i}\left(x^{*}\right)+{\left({\mu }_{m_i}^{*}\right)}_0}{-g_0^{m_i}\left(x^{*}\right)}}{\overline{g}}^{m_i}\left(x^{*}\right)\right).$$

For $V^{m_i}\in \partial {\mathrm{\Phi }}^{m_i}(0,-g^{m_i}\left(x^{*}\right)-{\mu }_{m_i}^{*})$, it has the following form:

$$V^{m_i}={\displaystyle \frac{1}{2}}\left(\begin{array}{ccc}0\hfill & 1\hfill & {\displaystyle \frac{-{\overline{g}}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}}{\parallel {\overline{g}}^{m_i}\left(x^{*}\right)\parallel }}\hfill \\ 0\hfill & {\displaystyle \frac{-{\overline{g}}^{m_i}\left(x^{*}\right)}{\parallel {\overline{g}}^{m_i}\left(x^{*}\right)\parallel }}\hfill & U^{m_i}\hfill \end{array}\right),$$

(47)

where

$$U^{m_i}=I_{m_i}+{\displaystyle \frac{-g_0^{m_i}\left(x^{*}\right)-{\left({\mu }_{m_i}^{*}\right)}_0}{-g_0^{m_i}\left(x^{*}\right)+{\left({\mu }_{m_i}^{*}\right)}_0}}{I}_{m_i}-{\displaystyle \frac{-g_0^{m_i}\left(x^{*}\right)-{\left({\mu }_{m_i}^{*}\right)}_0}{-g_0^{m_i}\left(x^{*}\right)+{\left({\mu }_{m_i}^{*}\right)}_0}}·{\displaystyle \frac{{\overline{g}}^{m_i}\left(x^{*}\right){\overline{g}}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}}{\parallel {\overline{g}}^{m_i}\left(x^{*}\right){\parallel }^2}}.$$

From (47), we obtain that

$$〈\left({(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x)}_0,-\overline{\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x});(-g^{m_i}\left(x^{*}\right),{\overline{g}}^{m_i}\left(x^{*}\right))\right)〉=0.$$

Thus, for $m_i\in S_2$, we have

$$\langle -\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x,{\mu }_{m_i}^{*}\rangle =0.$$

(c) If $m_i\in N_2$, we have $g^{m_i}\left(x^{*}\right)=0$ and ${\mu }_{m_i}^{*}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$. Then, $V^{m_i}\in \partial {\mathrm{\Phi }}^{m_i}(0,-g^{m_i}\left(x^{*}\right)-{\mu }_{m_i}^{*})$ can be expressed $V^{m_i}={\displaystyle \sum _{j=1}^p}{\alpha }_j^{m_i}{V}_j^{m_i}$, where ${\displaystyle \sum _{j=1}^p}{\alpha }_j^{m_i}=1$, ${\alpha }_j^{m_i}\ge 0$, $V_j^{m_i}\in {\partial }_B{\mathrm{\Phi }}^{m_i}(0,-g^{m_i}\left(x^{*}\right)-{\mu }_{m_i}^{*})$. Since there exists ${\alpha }_j^{m_i}\in [0,1]$ such that

$$V_j^{m_i}(0,-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x-\Delta \mu )={\displaystyle \frac{{\alpha }_j^{m_i}}{2}}\left(\begin{array}{cc}1\hfill & -{\displaystyle \frac{{\overline{\mu }}_{m_i}^{\mathrm{T}}}{\parallel {\overline{\mu }}_{m_i}\parallel }}\hfill \\ -{\displaystyle \frac{{\overline{\mu }}_{m_i}}{\parallel {\overline{\mu }}_{m_i}\parallel }}\hfill & -{\displaystyle \frac{{\overline{\mu }}_{m_i}{\overline{\mu }}_{m_i}^{\mathrm{T}}}{\parallel {\overline{\mu }}_{m_i}{\parallel }^2}}\hfill \end{array}\right)(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x-\Delta \mu ),$$

and thus,

$$V^{m_i}(0,-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x-\Delta \mu )=\sum _{j=1}^p{\alpha }_j^{m_i}{\displaystyle \frac{{\alpha }_j^{m_i}}{2}}\left(\begin{array}{cc}1\hfill & -{\displaystyle \frac{{\overline{\mu }}_{m_i}^{\mathrm{T}}}{\parallel {\overline{\mu }}_{m_i}\parallel }}\hfill \\ -{\displaystyle \frac{{\overline{\mu }}_{m_i}}{\parallel {\overline{\mu }}_{m_i}\parallel }}\hfill & -{\displaystyle \frac{{\overline{\mu }}_{m_i}{\overline{\mu }}_{m_i}^{\mathrm{T}}}{\parallel {\overline{\mu }}_{m_i}{\parallel }^2}}\hfill \end{array}\right)(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x-\Delta \mu ),$$

we can obtain

$$-{(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x)}_0{\overline{\mu }}_{m_i}={\mu }_0^{m_i}\overline{(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x)},$$

which implies that, for any $m_i\in N_2$,

$$-\mathcal{J}{g}^{m_i}\left(x^{*}\right)\Delta x\in \Re ({\mu }_0^{m_i},-{\overline{\mu }}^{m_i}).$$

Then, we can obtain that

$$\Delta x\in \mathrm{aff}\left(C\left(x^{*}\right)\right).$$

From the second and third equations of (46), we have

$$\begin{array}{c}0=\langle \Delta x,{\mathcal{J}}_x\mathcal{L}(x^{*},\mu )\Delta x+\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta \mu \rangle \hfill \\ =\langle \Delta x,{\mathcal{J}}_x\mathcal{L}(x^{*},\mu )\Delta x\rangle +\langle \Delta \mu ,\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta x\rangle .\hfill \end{array}$$

Combining with the last Equation (46), Proposition 1, and (37), the following is implied:

$$\langle \Delta x,{\mathcal{J}}_x\mathcal{L}(x^{*},{\mu }^{*})\Delta x\rangle +\langle \Delta x,\mathcal{H}(x^{*},{\mu }^{*})\Delta x\rangle \le 0.$$

(48)

Then, from the strong second-order sufficient condition (37) and (46), we conclude that $\Delta x=0$. Therefore, (46) reduces to

$$\left(\begin{array}{c}\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta \mu \hfill \\ V(0,\Delta \mu )\hfill \end{array}\right)=0.$$

(49)

From the second equation of (49), the index sets (20), Lemma 5, and B, we could easily obtain

$$\Delta {\mu }^{I^{*}}=0,\Delta {\mu }^{B^{*}}=0.$$

(50)

Then, it follows from the constraint nondegeneracy condition that there exists a vector $d\in {\Re }^n$ such that

$$-\mathcal{J}{g}^{Z^{*}}\left(x^{*}\right)d=-\Delta {\mu }^{Z^{*}}.$$

Then, we have

$$\begin{array}{c}\langle \Delta \mu ,\Delta \mu \rangle =\langle \Delta {\mu }^{Z^{*}},\Delta {\mu }^{Z^{*}}\rangle \hfill \\ =\langle \Delta {\mu }^{Z^{*}},\mathcal{J}{g}^{Z^{*}}\left(x^{*}\right)d\rangle \hfill \\ =\langle d,\mathcal{J}{g}^{Z^{*}}{\left(x^{*}\right)}^{\mathrm{T}}\Delta {\mu }^{Z^{*}}\rangle \hfill \\ =\langle d,\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta \mu \rangle \hfill \\ =0,\hfill \end{array}$$

which implies that

$$\Delta {\mu }^{Z^{*}}=0.$$

(51)

Together with (51), the conditions $\Delta \epsilon =0,\Delta x=0$, and (50) imply that W is nonsingular. This completes the proof. □

Corollary 1.

Let $(x^{*},{\mu }^{*})$ be a Karush–Kuhn–Tucker point of the SOCCVI problem (1). If the constraint nondegeneracy condition and the strong second-order sufficient condition (37) hold at $x^{*}$, then any element in $\partial Q(x^{*},{\mu }^{*})$ is nonsingular.

Proof. 

For any element $\tilde{W}\in \partial Q(x^{*},{\mu }^{*})$, we compute

$$\tilde{W}\left(\begin{array}{c}\Delta x\hfill \\ \Delta \mu \hfill \end{array}\right)=\left(\begin{array}{c}{\mathcal{J}}_x\mathcal{L}(x^{*},{\mu }^{*})\Delta x+\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta \mu \hfill \\ \mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta x+\tilde{V}(-\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta x-\Delta \mu )\hfill \end{array}\right),$$

(52)

where $\tilde{V}\in \partial \mathrm{\Pi }(-g\left(x^{*}\right)-{\mu }^{*})$. From Lemma 6, there exists $V\in \partial \mathrm{\Phi }(0,-g\left(x^{*}\right)-{\mu }^{*})$ such that

$$\tilde{V}(0,-\mathcal{J}g\left(x^{*}\right)\Delta x-\Delta \mu )=\tilde{V}(-\mathcal{J}g\left(x^{*}\right)\Delta x-\Delta \mu ).$$

Then, we obtain

$$\begin{array}{c}\left(\begin{array}{c}0\hfill \\ \tilde{W}\left(\begin{array}{c}\Delta x\hfill \\ \Delta \mu \hfill \end{array}\right)\hfill \end{array}\right)=\left(\begin{array}{c}0\hfill \\ {\mathcal{J}}_x\mathcal{L}(x^{*},{\mu }^{*})\Delta x+\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta \mu \hfill \\ \mathcal{J}g\left(x^{*}\right)\Delta x+V(0,-\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}\Delta x-\Delta \mu )\hfill \end{array}\right)\hfill \\ =\tilde{W}\left(\begin{array}{c}0\hfill \\ \Delta x\hfill \\ \Delta \mu \hfill \end{array}\right),\hfill \end{array}$$

where $\tilde{W}\in \partial E(x^{*},{\mu }^{*})$ is defined by (46).

Under the constraint nondegeneracy condition and the strong second-order sufficient condition (37), it follows from Theorem 2 that

$$\tilde{W}\left(\begin{array}{c}0\hfill \\ \Delta x\hfill \\ \Delta \mu \hfill \end{array}\right)=0,$$

which implies that $\Delta x=0,\Delta \mu =0$; that is, $\tilde{W}$ is nonsingular. This completes the proof. □

The next theorem demonstrates the relationship between the strongly regular solution and the strong second-order sufficient condition.

Theorem 3.

Let $x^{*}$ be a locally optimal solution of the SOCCVI problem (1), and ${\mu }^{*}$ is the corresponding Lagrange multiplier. Then, $(x^{*},{\mu }^{*})$ is the strongly regular solution of the generalized Equation (41) if and only if $x^{*}$ is nondegenerate and the following strongly second-order sufficient condition holds at $(x^{*},{\mu }^{*})$:

$$Q_0\left(d\right):={\mathcal{J}}_x\mathcal{L}(x^{*},{\mu }^{*})(d,d)+d^{\mathrm{T}}\mathcal{H}(x^{*},{\mu }^{*})d>0,\forall d\in \mathrm{aff}\left(\mathcal{C}\left(x^{*}\right)\right)\setminus \left\{0\right\}.$$

(53)

Proof. 

From Theorem 5.25 in Bonnans and Ramirez [6], we know that $x^{*}$ is nondegenerate, and the uniform second-order growth condition holds at $x^{*}$ if and only if $x^{*}$ is a strongly regular solution of the generalized Equation (41). Hence, under the nondegenerate condition, we only prove that the strong second-order sufficient condition is equivalent to the uniform second-order growth condition.

If $x^{*}$ is nondegenerate, then the affine space of the critical cone at $x^{*}$ is as follows:

$$\mathrm{aff}\left(\mathcal{C}\left(x^{*}\right)\right)=\left\{\begin{array}{cc}d\in {\Re }^n,\hfill & i=1,\cdots ,p,\hfill \\ -\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}d=0\hfill & {\mu }^{m_i}\in \mathrm{int}{\mathcal{K}}^{m_i},\hfill \\ -\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}d\in \Re ({\mu }_0^{m_i},-{\overline{\mu }}^{m_i})\hfill & {\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\},-g^{m_i}\left(x^{*}\right)=0,\hfill \\ -\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}d{\mu }^{m_i}=0\hfill & {\mu }^{m_i},-g^{m_i}\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}.\hfill \end{array}\right.$$

(54)

The uniform second-order growth condition implies that the strong second-order sufficient condition holds. Consider that the vector space E is defined by

$$E=\left\{\begin{array}{cc}d\in {\Re }^n,\hfill & i=1,\cdots ,p,\hfill \\ -\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}d=0,\hfill & {\mu }^{m_i}\in \mathrm{int}{\mathcal{K}}^{m_i},\hfill \\ -\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}d{\mu }^{m_i}=0\hfill & {\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}.\hfill \end{array}\right.$$

(55)

Obviously, we obtain that aff $\left(\mathcal{C}\left(x^{*}\right)\right)\subset E$.

Next, we consider the perturbation problem of the second-order cone constrained optimization (SOCCOP) problem (23).

$$\begin{array}{c}\min f(x,u)\hfill \\ \mathrm{s}.\mathrm{t}.-g^{m_i}(x,u):=-g^{m_i}\left(x\right)+u{\delta }^{m_i}\in {\mathcal{K}}^{m_i}\hfill \end{array}$$

(56)

where

$${\delta }^{m_i}=\left\{\begin{array}{cc}{e}_1^{m_i},\hfill & {\mu }^{m_i}=0,\hfill \\ (-{\mu }_0^{m_i},{\overline{\mu }}^{m_i}),\hfill & -g^{m_i}\left(x^{*}\right)=0,{\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(57)

$e_1^{m_i}$ denotes the first element of the natural basis of ${\mathcal{K}}^{m_i}$, and $u>0$ is the perturbed parameter. In problem (56), $x^{*}$ is still a locally optimal solution with the same Lagrange multiplier ${\mu }^{*}$ such that a bigger critical cone is equal to E.

Now, we prove that the strong second-order sufficient condition of the perturbation problem of (56) can be obtained when $-g^{m_i}\left(x^{*}\right)=0$, ${\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$.

Define

$$C:=\left\{1\le i\le p|-g^{m_i}\left(x^{*}\right)=0,{\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}\right\}.$$

Let $\mathcal{H}(x^{*},{\mu }^{m_i},u)$ denote the matrices in the expression of the strong second-order sufficient condition.

For all $m_i\notin C$, we obtain that $\mathcal{H}(x^{*},{\mu }^{m_i},u)=\mathcal{H}(x^{*},{\mu }^{m_i})$; however, for all $m_i\in C$, we obtain that

$${\mathcal{H}}^{m_i}(x^{*},{\mu }^{m_i},u)={\displaystyle \frac{1}{u}}\widehat{\mathcal{H}}(x^{*},{\mu }^{m_i}),$$

(58)

where ${\widehat{\mathcal{H}}}^{m_i}(x^{*},{\mu }^{m_i})=\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}\left(\begin{array}{cc}1\hfill & 0^{\mathrm{T}}\hfill \\ 0\hfill & -I_{m_i}\hfill \end{array}\right)(-\mathcal{J}{g}^{m_i}\left(x^{*}\right))$. We set

$$Q_1\left(d\right):={\displaystyle \sum _{m_i\in C}}{d}^{\mathrm{T}}\widehat{H}(x^{*},{\mu }^{m_i})d={\displaystyle \sum _{m_i\in C}}\left(\parallel \mathcal{J}{\overline{g}}^{m_i}\left(x^{*}\right){d\parallel }^2-{\left(\mathcal{J}{g}_0^{m_i}\left(x^{*}\right)d\right)}^2\right).$$

(59)

If $d\in E$, then we have $-\mathcal{J}{g}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}d{\mu }^{m_i}=0$, ${\mu }_0^{m_i}=\parallel {\overline{\mu }}^{m_i}\parallel$. Hence, we obtain that

$$|-\mathcal{J}{g}_0^{m_i}\left(x^{*}\right)d|={\displaystyle \frac{|-\mathcal{J}{\overline{g}}^{m_i}\left(x^{*}\right)d{\overline{\mu }}^{m_i}|}{{\mu }_0^{m_i}}}\le \parallel -\mathcal{J}{\overline{g}}^{m_i}\left(x^{*}\right)d\parallel$$

(60)

is equivalent to $-\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}d\in \Re ({\mu }_0^{m_i},-{\overline{\mu }}^{m_i})$. Combining with (59), we can obtain $Q_1\left(d\right)\ge 0$ for all $d\in E$ and $Q_1\left(d\right)=0$ for all $d\in$ aff$\left(\mathcal{C}\left(x^{*}\right)\right)$.

For sufficiently small u, the uniform second-order growth of the perturbation problem (56) implies that

$$Q_0\left(d\right)+{\displaystyle \frac{1}{u}}{Q}_1\left(d\right)>0,d\in E\setminus \left\{0\right\}.$$

(61)

Since $Q_1\left(d\right)=0$ for $d\in E$, we have $Q_0\left(d\right)>0$ from (61). Thus, the strongly second-order sufficient condition (53) holds at $(x^{*},{\mu }^{*})$. For any $x_n\to x^{*}$ and $u_n\to 0$, there exist $d_n\in {\Re }^n,d_n\ne 0$ and $d_n\to 0$ such that $x_n+d_n$ is a feasible point of the perturbation problem (44); that is, $-g(x_n+d_n,u_n)\in \mathcal{K}$ and

$$f(x_n+d_n,u_n)\le f(x_n,u_n)+o(\parallel d_n{\parallel }^2).$$

(62)

For sufficiently large n, there exists a unique Lagrange multiplier ${\mu }_n$ of the stationary point $x_n$ of the perturbation problem (44). Since $x_n\to x^{*}$, we have ${\mu }_n\to {\mu }^{*}$. We assume that $d_n/\parallel d_n\parallel$ converge to some $d^{*}\ne 0$. Let us check that $d^{*}\in$ aff$\left(\mathcal{C}\left(x^{*}\right)\right)$. For $-g^{m_i}(x_n+d_n,u_n)\in {\mathcal{K}}^{m_i}$, we know that

$$-g^{m_i}(x_n+d_n,u_n)=-g^{m_i}(x_n,u_n)-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n+\mathrm{o}(\parallel d_n\parallel ){⪰}_{{\mathcal{K}}^{m_i}}0.$$

(63)

It follows from $-g^{m_i}(x_n,u_n){\mu }_n^{m_i}=0$ that

$${\mu }_n^{m_i}(-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n)+o(\parallel d_n\parallel )\ge 0,$$

(64)

Dividing it by $\parallel d_n\parallel$ implies that

$${\displaystyle \frac{{\mu }_n^{m_i}(-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n)+o(\parallel d_n\parallel )}{\parallel d_n\parallel }}\ge 0.$$

Let $n\to \infty$ in the above inequality, and we obtain that ${\mu }^{m_i}(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d^{*})\ge 0$. Combining the first equation of the KKT condition and $n\to \infty$ in (62), we obtain that

$$\nabla f\left(x^{*}\right)d^{*}=\mu (-\mathcal{J}g\left(x^{*}\right)d^{*})={\displaystyle \sum _{i=1}^p}{\mu }^{m_i}(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d^{*})\le 0.$$

Hence, we have

$$-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d^{*}{\mu }^{m_i}=0,(i=1,\cdots ,p).$$

(65)

According to the characterization of aff$\left(\mathcal{C}\left(x^{*}\right)\right)$, we consider the following three cases:

Case I: For ${\mu }^{m_i}$, $-g^{m_i}\left(x^{*}\right)\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$, we obtain that (65) holds.

Case II: For ${\mu }^{m_i}\in \mathrm{int}{\mathcal{K}}^{m_i},-g^{m_i}\left(x^{*}\right)=0$, it follows from ${\mu }_n^{m_i}\to {\mu }^{m_i}(n\to \infty )$ that ${\mu }_n^{m_i}\in \mathrm{int}{\mathcal{K}}^{m_i}$ and $-g^{m_i}(x_n,u_n)=0$ for sufficiently large n. There exists $\epsilon >0$ such that ${\mu }^{m_i}+2\epsilon B\subset {\mathcal{K}}^{m_i}$. Then, for any unit vector z, ${\mu }_n^{m_i}+\epsilon z\subset {\mathcal{K}}^{m_i}$ for sufficiently large n, we compute the scalar product of ${\mu }_n^{m_i}+\epsilon z$ and (64). Moreover, let $n\to \infty$, and we obtain that

$$({\mu }^{m_i}+\epsilon z)(-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d^{*})\ge 0.$$

The above inequality holds for any z. Using (65), we have

$$-\mathcal{J}{g}^{m_i}\left(x^{*}\right)d^{*}=0.$$

(66)

Case III: For ${\mu }^{m_i}\in \mathrm{bd}{\mathcal{K}}^{m_i}\setminus \left\{0\right\}$ and $-g^{m_i}\left(x^{*}\right)=0$, we will prove that $-g^{m_i}\left(x^{*}\right)\in \Re ({\mu }_0^{m_i},-{\overline{\mu }}^{m_i})$.

By (60) and (65), we obtain that

$$|-\mathcal{J}{g}_0^{m_i}\left(x^{*}\right)d^{*}{|}^2\ge {\parallel -\mathcal{J}{\overline{g}}^{m_i}{\left(x^{*}\right)}^{\mathrm{T}}{d}^{*}\parallel }^2.$$

(67)

By Taylor expansion in (62), we deduce that

$${\nabla }_{x}f(x_n,u_n)d_n\le \mathrm{O}(\parallel d_n{\parallel }^2).$$

Combining with the first-order optimality conditions of the perturbation problem (44), we deduce that

$${\displaystyle \sum _{i=1}^p}{\mu }_n^{m_i}(-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n)={\mu }_n(-\mathcal{J}g(x_n,u_n)d_n)={\nabla }_{x}f(x_n,u_n)d_n\le \mathrm{O}(\parallel d_n{\parallel }^2).$$

(68)

Similarly to (63) and (64), we can obtain that

$$-g^{m_i}(x_n+d_n,u_n)=-g^{m_i}(x_n,u_n)-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n+\mathrm{O}(\parallel d_n{\parallel }^2)\in {\mathcal{K}}^{m_i}.$$

(69)

Since ${\mathcal{K}}^{m_i}$ is self-dual, ${\mu }_n$ is bounded, and ${\mu }_n^{m_i}(-g^{m_i}(x_n,u_n))=0$, we deduce that

$$-{\mu }_n^{m_i}(-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n)\le \mathrm{O}(\parallel d_n{\parallel }^2).$$

In view of (68), we have

$$|{\mu }_n^{m_i}(-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n)|=\mathrm{O}(\parallel d_n{\parallel }^2).$$

(70)

Since ${\mu }_n^{m_i}\to {\mu }^{m_i}\ne 0(n\to \infty )$, we have $-g^{m_i}(x_n,u_n)={\alpha }_n\left({\left({\mu }_n^{m_i}\right)}_0;-{\overline{\mu }}_n^{m_i})\right)$ for some sequence ${\alpha }_n>0$ and ${\alpha }_n\to 0(n\to \infty )$. Substituting (69) into the inequality $|-g_0^{m_i}(x_n+d_n,u_n){|}^2\ge {\parallel -{\overline{g}}^{m_i}(x_n+d_n,u_n)\parallel }^2$ and using $|-g_0^{m_i}(x_n,u_n){|}^2={\parallel -{\overline{g}}^{m_i}(x_n,u_n)\parallel }^2$, we obtain that

$$\begin{array}{c}|-\mathcal{J}{g}_0^{m_i}(x_n,u_n)d_n{|}^2-{\parallel -\mathcal{J}{\overline{g}}^{m_i}(x_n,u_n)d_n\parallel }^2-2{\alpha }_n\mathcal{J}{\overline{g}}^{m_i}(x_n,u_n)d_n{\mu }_n^{m_i}\hfill \\ \ge o(\parallel d_n\parallel (\parallel -\mathcal{J}{g}^{m_i}(x_n,u_n)d_n\parallel +\parallel d_n\parallel \left)\right)+\mathrm{O}({\alpha }_n\parallel d_n{\parallel }^2).\hfill \end{array}$$

(71)

Using (71) and $\parallel -\mathcal{J}{\overline{g}}^{m_i}(x_n,u_n)d_n\parallel =\mathrm{O}(\parallel d_n\parallel )$, we obtain that

$$|-\mathcal{J}{g}_0^{m_i}\left(x^{*}\right)d^{*}{|}^2\ge {\parallel -\mathcal{J}{\overline{g}}^{m_i}\left(x^{*}\right)d^{*}\parallel }^2.$$

From the above three cases, we prove that $0\ne d^{*}\in$aff$\left(C\left(x^{*}\right)\right)$.

Let

$$D:=\left\{1\le i\le p:-g^{m_i}\left(x^{*}\right)\ne 0\ne {\mu }^{m_i}\right\}$$

(72)

and $\phi (-g^{m_i}(x,u)):=\parallel -{\overline{g}}^{m_i}(x,u)\parallel -{(-g^{m_i}(x,u))}_0$; then, $-g^{m_i}(x,u)\in {\mathcal{K}}^{m_i}$ means that $\phi (-g^{m_i}(x,u))\le 0$. We denote by ${\mu }_n$ the Lagrange multiplier associated with $x_n$, and we know that

$${\displaystyle \sum _{m_i\notin D}}{\mu }_n^{m_i}(-g^{m_i}(x,u))+{\displaystyle \sum _{m_i\in D}}{\left({\mu }_n\right)}_0^{m_i}\phi (-g^{m_i}(x,u))\ge 0.$$

(73)

Substituting $(x_n+d_n,u_n)$ in (73) and using the complementarity conditions at $(x_n,u_n)$, we deduce that

$$\begin{array}{c}{\displaystyle \sum _{m_i\notin D}}{\mu }_n^{m_i}\left((-g^{m_i}(x_n+d_n,u_n)-(-g^{m_i}(x_n,u_n)\right)\hfill \\ +{\displaystyle \sum _{m_i\in D}}{\left({\mu }_n\right)}_0^{m_i}(\phi (-g^{m_i}(x_n+d_n,u_n))-\phi (-g^{m_i}(x_n,u_n))\ge 0.\hfill \end{array}$$

(74)

Adding (62) and (74), we can the following:

$$\begin{array}{c}{▽}_{xx}^{2}f(x_n,u_n)(d_n,d_n)-{\displaystyle \sum _{m_i\notin D}}{\mu }_n^{m_i}{▽}_{xx}^2(-g(x_n,u_n))(d_n,d_n)\\ -{\displaystyle \sum _{m_i\in D}}{\left({\mu }_n\right)}_0^{m_i}(\phi (-g^{m_i}(x_n,u_n))(-\mathcal{J}{\overline{g}}^{m_i}(x_n,u_n)d_n,-\mathcal{J}{g}^{m_i}(x_n,u_n)d_n)\le \mathrm{o}(\parallel d_n{\parallel }^2).\end{array}$$

(75)

Similarly to Lemma 27 in Bonnans and Ramirez [6], let $n\to \infty$ in (75), and we can obtain that $Q_0\left(d^{*}\right)\le 0$, which is a contradiction with the strongly second-order sufficiently condition, since $d^{*}\in$aff$\left(\mathcal{C}\left(x^{*}\right)\right)\setminus \left\{0\right\}$. Hence, the uniform second-order growth condition holds. This completes the proof. □

From the definition of the function Q in (38), we know that the Karush–Kuhn–Tucker point $(x^{*},{\mu }^{*})$ of the SOCCVI problem (1) satisfies $Q(x^{*},{\mu }^{*})=0$. The directional derivative of Q along $\delta :=({\delta }_1,{\delta }_2)\in {\Re }^n\times {\Re }^m$ is as follows:

$$Q^{\prime }(x^{*},{\mu }^{*};\delta )=\left(\begin{array}{c}{▽}_x\mathcal{L}(x^{*},{\mu }^{*}){\delta }_1+\mathcal{J}g{\left(x^{*}\right)}^{\mathrm{T}}{\delta }_2\hfill \\ \mathcal{J}g\left(x^{*}\right){\delta }_1-{\mathrm{\Pi }}_{\mathcal{K}}^{\prime }(-g\left(x^{*}\right)-{\mu }^{*},-\mathcal{J}g\left(x^{*}\right){\delta }_1-{\delta }_2)\hfill \end{array}\right)=:\mathrm{\Psi }\left(\delta \right).$$

(76)

Since $\mathrm{\Psi }(·)$ is Lipschitz continuous, ${\partial }_B\mathrm{\Psi }\left(0\right)$ is well defined.

Proposition 2.

Let $u^{*}=-g\left(x^{*}\right)-{\mu }^{*}$ and $\mathrm{\Psi }\left(\delta \right)$ be defined by (76); then,

$${\partial }_B\mathrm{\Psi }\left(0\right)={\partial }_{B}Q(x^{*},{\mu }^{*}).$$

(77)

Proof. 

Let $\mathrm{\Theta }:{\Re }^n\times {\Re }^m\to {\Re }^m$ be as follows:

$$\mathrm{\Theta }\left(\delta \right):={\mathrm{\Pi }}_{\mathcal{K}}^{\prime }(-g\left(x^{*}\right)-{\mu }^{*},-\mathcal{J}g\left(x^{*}\right){\delta }_1+{\delta }_2)={\mathrm{\Pi }}_{\mathcal{K}}^{\prime }(u^{*},\gamma \left(\delta \right)),$$

where $\delta :=({\delta }_1,{\delta }_2)\in {\Re }^n\times {\Re }^m$ and $\gamma \left(\delta \right):=-\mathcal{J}g\left(x^{*}\right){\delta }_1+{\delta }_2$.

Let $\mathrm{\Gamma }(·):={\mathrm{\Pi }}_{\mathcal{K}}^{\prime }(u,·)$. From Lemma 14 in Pang et al. [18], ${\partial }_B\mathrm{\Gamma }\left(0\right)={\partial }_B{\mathrm{\Pi }}_{\mathcal{K}}\left(u\right)$ holds. We have

$${\partial }_B\mathrm{\Theta }\left(0\right)={\partial }_B{\mathrm{\Pi }}_{\mathcal{K}}\left(u\right)J_{\delta }\gamma \left(0\right),$$

and together with (76) and Lemma 4, we obtain that (76) holds. This completes the proof. □

According to Clarke’s inverse theorem in Clarke [19,20], we can obtain that any element in $\partial Q(x^{*},{\mu }^{*})$ is nonsingular if and only if the function Q is a locally Lipschitz homeomorphism near the Karush–Kuhn–Tucker point $(x^{*},{\mu }^{*})$ in the following theorem:

Theorem 4.

If $\partial Q\left(x^{*}\right)$ is nonsingular, then there exist the neighborhood U of $x^{*}$ and V of $Q\left(x^{*}\right)$, as well as the Lipschtiz function $G:V\to {\Re }^n$ such that

$\left(i\right)G\left(Q\right(u\left)\right)=U,\forall u\in U.$

$\left(ii\right)Q\left(G\right(v\left)\right)=V,\forall v\in V.$

The next lemma demonstrates that the relationship between the strongly regular solution and the function Q is that Q is a locally Lipschitz homeomorphism near the Karush–Kuhn–Tucker point $(x^{*},{\mu }^{*})$.

Lemma 7.

Suppose that $\mathcal{Z}={\Re }^n\times {\Re }^m$, $Q:\mathcal{Z}\to \mathcal{Z}$ is defined by (38), and ${\mathcal{Z}}^{*}=(x^{*},{\mu }^{*})$ is a KKT point of the SOCCVI problem (1). Then, Q is a locally Lipschitz homeomorphism at $\mathcal{Z}$ if and only if $\mathcal{Z}$ is a strongly regular solution of the generalized Equation (41).

Proof. 

Assuming that Q is a locally Lipschitz homeomorphism at ${\mathcal{Z}}^{*}$, there exists an open neighborhood $\mathcal{V}$ of ${\mathcal{Z}}^{*}$ which satisfies the condition that $Q\left(\mathcal{V}\right)$ is an open neighborhood of the origin $0\in \mathcal{Z}$. For any $\widehat{\delta }\in Q\left(\mathcal{V}\right)$, the equation $Q\left(\mathcal{V}\right)=\widehat{\delta }$ has a unique solution ${\widehat{\mathcal{Z}}}_{\mathcal{V}}$ in $\mathcal{V}$, and ${\widehat{\mathcal{Z}}}_{\mathcal{V}}:Q\left(\mathcal{V}\right)\to \mathcal{V}$ is Lipschitz continuous.

For any $\delta =({\delta }_1,{\delta }_2)\in \mathcal{M}\equiv {\displaystyle \frac{1}{2}}Q\left(\mathcal{V}\right)$, let $Z\left(\delta \right)=\left(x\right(\delta ),\mu (\delta \left)\right)$ be a solution of the generalized Equation (41). Denote $\delta =({\delta }_1,{\delta }_2)\in {\Re }^n\times {\Re }^m$, and then we have

$$\delta \in \left(\begin{array}{c}\mathcal{L}(x,u)\hfill \\ -g\left(x\right)\hfill \end{array}\right)+\left(\begin{array}{c}{N}_{{\Re }^n}\left(x\right)\hfill \\ N_{\mathcal{K}}\left(\mu \right)\hfill \end{array}\right),$$

which implies that

$$\left(\begin{array}{c}F\left(x\left(\delta \right)\right)+\mathcal{J}g{\left(x\left(\delta \right)\right)}^{\mathrm{T}}(\mu \left(\delta \right)+{\delta }_2)\hfill \\ \mu \left(\delta \right)+{\delta }_2-{\mathrm{\Pi }}_{\mathcal{K}}(\mu \left(\delta \right)+{\delta }_2-g\left(x\left(\delta \right)\right))\hfill \end{array}\right)=\left(\begin{array}{c}{\delta }_1\hfill \\ {\delta }_2\hfill \end{array}\right),$$

that is,

$$Q(x\left(\delta \right),\mu \left(\delta \right)+{\delta }_2)=\left(\begin{array}{c}{\delta }_1-\mathcal{J}g{\left(x\left(\delta \right)\right)}^{\mathrm{T}}{\delta }_2\\ {\delta }_2\end{array}\right).$$

Then, $\mathcal{Z}\left(\delta \right)$ uniquely exists in $\mathcal{V}$ and

$$\mathcal{Z}\left(\delta \right)={\widehat{\mathcal{Z}}}_{\mathcal{V}}({\delta }_1-\mathcal{J}g{\left(x\left(\delta \right)\right)}^{\mathrm{T}}{\delta }_2,{\delta }_2)-\left(\begin{array}{c}0\hfill \\ {\delta }_2\hfill \end{array}\right).$$

Thus, $\mathcal{Z}(·)$ is Lipschitz continuous on $\mathcal{M}$.

Assuming that ${\mathcal{Z}}^{*}$ is a strongly regular solution of the generalized Equation (41), there exist neighborhoods $\mathcal{M}$ of the origin $0\in \mathcal{Z}$ and $\mathcal{V}$ of ${\mathcal{Z}}^{*}$, as well as a locally Lipschitz function ${\mathcal{Z}}_{\mathcal{V}}:\mathcal{M}\to \mathcal{V}$, which satisfy the condition that ${\mathcal{Z}}_{\mathcal{V}}\left(\delta \right)$ is the unique solution in $\mathcal{V}$ for any $\delta \in \mathcal{M}$. According to the above proof, we can conclude that $Q\left(\mathcal{Z}\right)=\widehat{\delta }$ has a unique solution for any $\widehat{\delta }=({\widehat{\delta }}_1,{\widehat{\delta }}_2)\in \left({\displaystyle \frac{1}{2}}\mathcal{M}\right)\cap ({\Re }^n\times {\Re }^m)$ and

$$Q(x\left(\widehat{\delta }\right),\mu \left(\widehat{\delta }\right)-{\widehat{\delta }}_2)=\left(\begin{array}{c}{\widehat{\delta }}_1+\mathcal{J}g{\left(x\left(\delta \right)\right)}^{\mathrm{T}}{\widehat{\delta }}_2\\ -{\widehat{\delta }}_2\end{array}\right),$$

where $\widehat{\mathcal{Z}}\left(\widehat{\delta }\right)\in \mathcal{V}$ and

$$\widehat{\mathcal{Z}}\left(\widehat{\delta }\right)={\mathcal{Z}}_{\mathcal{V}}({\widehat{\delta }}_1-\mathcal{J}g{\left(x\left(\delta \right)\right)}^{\mathrm{T}}widehat{\delta }_2,-{\widehat{\delta }}_2)+\left(\begin{array}{c}0\hfill \\ {\widehat{\delta }}_2\hfill \end{array}\right),$$

which implies that $\widehat{\mathcal{Z}}(·)$ is Lipschitz continuous on ${\displaystyle \frac{1}{2}}\mathcal{M}$. Thus, Q is Lipschitz homeomorphism at ${\mathcal{Z}}^{*}$. This completes the proof. □

Now, we demonstrate the stability analysis theorem of the SOCCVI problem (1).

Theorem 5.

Let $x^{*}$ be a locally optimal solution of the SOCCVI problem (1), suppose that Robinson’s CQ holds at $x^{*}$, and let ${\mu }^{*}$ be a Lagrangian multiplier associated with $x^{*}$. Then, the following conclusions are equivalent to each other:

(a) 

The strong second-order sufficient condition (37) holds at $x^{*}$, and $x^{*}$ is constraint nondegenerate.

(b) 

Any element in $\partial E(0,x^{*},{\mu }^{*})$ is nonsingular.

(c) 

Any element in $\partial Q(x^{*},{\mu }^{*})$ is nonsingular.

(d) 

The KKT point $(x^{*},{\mu }^{*})$ is a strongly regular solution of the generalized Equation (40).

(e) 

The point $x^{*}$ is nondegenerate, and the uniform second-order growth condition holds at $x^{*}$.

(f) 

The point $x^{*}$ is strongly stable, and $x^{*}$ is constraint nondegenerate.

(g) 

The function Q is a locally Lipschitz homeomorphism at the KKT point $(x^{*},{\mu }^{*})$.

(h) 

Any element in $\partial \mathrm{\Psi }\left(0\right)$ is nonsingular.

Proof. 

Obviously, it follows from Theorem 2 and Corollary 1 that (a)⇒(b)⇒(c). According to Theorem 5.24 and Theorem 5.35 in Bonnans and Shapiro [14], we can obtain that (d)⇔(e)⇔(f). From Theorem 3, we have (a)⇔(d). Proposition 2 implies that (h)⇔(c). From Clarke’s inverse theorem, we have (c)⇔(g). From Lemma 7, we have (d)⇔(g). This completes the proof. □

5. Conclusions

In this paper, we obtain the optimality conditions for the SOCCVI problem (1). Moreover, we prove that the strong second-order sufficient condition under the nondegenerate condition is equivalent to the nonsingularlity of Clarke’s generalized Jacobian, and it is equivalent to the strong regularity of the KKT point. Then, the uniform second-order growth condition and the strong stability under the nondegenerate contrition are equivalent to each other. We demonstrate the equivalence between the strong stability of the KKT point and the local Lipschitz homeomorphism near the KKT point. Finally, all conditions are proved to be equivalent to each other. The above results for the SOCCVI problem (1) will be a supplement for the theoretical analysis and the convergence of algorithms to solve the SOCCVI problem (1). The works in [2,21,22] have proposed different solution methods for the second-order cone programming and the second-order cone constrained variational inequality problems. The theoretical results established in this study offer crucial support for the development of numerical algorithms for solving the SOCCVI problem (1).