Characterization of the Convergence Rate of the Augmented Lagrange for the Nonlinear Semidefinite Optimization Problem

Abstract

The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector $(\pi ,Y)$ and the penalty parameter $\sigma$ are chosen such that $\sigma$ is larger than a threshold ${\sigma }^{*}>0$ and the ratio $\parallel (\pi ,Y)-({\pi }^{*},Y^{*})\parallel /\sigma$ is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to $\parallel (\pi ,Y)-({\pi }^{*},Y^{*})\parallel$ and the ratio constant is proportional to $1/\sigma$, where $({\pi }^{*},Y^{*})$ is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q-linear convergence rate when the sequence of penalty parameters $\left\{{\sigma }_k\right\}$ has an upper bound and the convergence rate is superlinear when $\left\{{\sigma }_k\right\}$ is increasing to infinity.

1. Introduction

Consider the following nonlinear semidefinite optimization problem

$$\begin{array}{cc}min\hfill & f\left(x\right)\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & h\left(x\right)=0,\hfill \\ \hfill & g\left(x\right)⪯0,\hfill \end{array}$$

(1)

where function $f:{\mathbb{R}}^n\to \mathbb{R}$, mapping $h:{\mathbb{R}}^n\to {\mathbb{R}}^q$ and mapping $g:{\mathbb{R}}^n\to {\mathbb{S}}^p$ are assumed to be twice continuously differentiable in a neighborhood of a given feasible point $\overline{x}\in {\mathbb{R}}^n$, where ${\mathbb{S}}^p$ is the space of $p\times p$ real symmetric matrices.

The augmented Lagrange method was initiated by Powell [1] and Hestenes [2] for solving equality constrained nonlinear programming problems and was extended by Rockafellar [3] to inequality constrained optimization problems. For convex programming, Rockafellar [3,4] adopted the augmented Lagrange function for establishing a saddle point theorem and demonstrated the global convergence of the augmented Lagrange method when the penalty parameter is chosen as an arbitrary positive number. Rockafellar [5] gave a deep study of the augmented Lagrange method for convex optimization.

The study of local convergence properties for the augmented Lagrange method is fairly comprehensive. For the optimization problems with equality constraints, Powell [1] proved the local linear convergence of the augmented Lagrange method to a local minimum point when the second-order sufficient condition and the linear independence constraint qualification are satisfied. This result was extended by Bertsekas ([6], Chapter 3) to the optimization problem with inequality constraints under the strict complementarity condition, in which the linear rate constant is proportional to $1/\sigma$. If the strict complementarity condition is not satisfied, Ito and Kunisch [7], Conn et al. [8] and Contesse-Becker [9] proved that the augmented Lagrange method has linear convergence rate.

The Lagrange function for Problem (1) can be written as

$$\mathbb{L}(x,\pi ,Y)=f\left(x\right)+{\pi }^{T}h\left(x\right)+\langle Y,g\left(x\right)\rangle \mathrm{where}(x,\pi ,Y)\in {\mathbb{R}}^n\times {\mathbb{R}}^q\times {\mathbb{S}}^p.$$

(2)

The augmented Lagrange function for (1) is defined by

$${\displaystyle {\mathbb{L}}_{\sigma }(x,\pi ,Y)=f\left(x\right)+\langle \pi ,h\left(x\right)\rangle +\frac{\sigma }{2}{\parallel h\left(x\right)\parallel }^2+\frac{1}{2\sigma }[\parallel {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}{(Y+\sigma g\left(x\right))\parallel }_F^2-{\parallel Y\parallel }_F^2],}$$

(3)

where ${\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}$ denotes the projection operator to ${\mathbb{S}}_{+}^p$. The augmented Lagrange method for Problem (1) can be expressed in the following form:

Step 0

Given parameter ${\sigma }_0>0$, initial point $x^0\in {\mathbb{R}}^n$, initial multiplier $({\pi }^0,Y^0)\in {\mathbb{R}}^q\times {\mathbb{S}}_{+}^p$ and $k=0$.

Step 1

If

$$\parallel {\nabla }_x\mathbb{L}(x^k,{\pi }^k,Y^k)\parallel +\parallel Y^k-{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y^k+g\left(x^k\right))\parallel +\parallel h\left(x^k\right)\parallel =0,$$

then stop and $(x^k,{\pi }^k,Y^k)$ is a Karush–Kuhn–Tucker (KKT) pair.

Step 2

Solve the following problem

$$x^{k+1}\in \mathrm{argmin}{\mathbb{L}}_{{\sigma }_k}(x,{\pi }^k,Y^k)$$

and calculate

$$Y^{k+1}={\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y^k+{\sigma }_{k}g\left(x^{k+1}\right))\mathrm{and}{\pi }^{k+1}={\pi }^k+{\sigma }_{k}h\left(x^{k+1}\right).$$

Step 3

Update ${\sigma }_{k+1}$, set $k+1$ to k, and go to Step 1.

For nonlinear semidefinite optimization problem, in the appendix of Sun et al. [10], they used a direct way to derive the linear rate of convergence when the strict complementarity condition holds. However, this result on the rate of convergence of the augmented Lagrange method obtained in [10] has the possibility for improvement. For example, can we obtain a similar result to ([6], Chapter 3) for equality constrained optimization problems when $\parallel ({\pi }^0,Y^0)-({\pi }^{*},Y^{*})\parallel /\sigma$ is very small? How can we characterize the rate constant of the local linear convergence of the augmented Lagrangian method? In this paper, we will give positive answers to these two questions.

It should be noted that there are a lot of important applications for augmented Lagrangian methods in different types of optimization problems; for examples, see [11,12,13].

The paper is organized as follows. In the next section, we develop properties of the augmented Lagrange function under the Jacobian uniqueness conditions for the semidefinite optimization problem (1), which will be required to prove results on the convergence rate of the augmented Lagrange method. In Section 3, we demonstrate the linear rate of convergence of the augmented Lagrangian method for the semidefinite optimization problem when the Jacobian uniqueness conditions are satisfied. In Section 4, we establish the asymptotical convergence rate of Lagrange multipliers, which shows that the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method is convergent to the optimal Lagrange multiplier superlinearly when the sequence $\left\{{\sigma }_k\right\}$ is increasing to ∞. Finally, we draw a conclusion in Section 5.

We list two technical results at the end of this section, which will be used in developing properties of the augmented Lagrange function for proving the main theorem about the convergence rate of the ALM. The first technical result is a variant of [14] and the second result is an implicit function theorem from page 12 of Bertsekas [6].

Lemma 1. 

Let X and Y be two finite dimensional Hilbert spaces and $\psi :X\to \mathbb{R}$ be continuous and positive homogeneous of degree 2, namely

$$\psi \left(tw\right)=t^2\psi \left(w\right)\forall t\ge 0 and w\in X.$$

Suppose that there exists a real number ${\kappa }_0>0$ such that $\psi \left(w\right)\ge {\kappa }_0{\parallel w\parallel }^2$ for any w satisfying $\mathcal{L}w=0$, where $\mathcal{L}:X\mapsto Y$ is a linear mapping. Then, there exist positive real numbers $\underline{\kappa }\in (0,{\kappa }_0]$ and $c_0>0$ such that

$$\psi \left(w\right)+c_0\langle \mathcal{L}w,\mathcal{L}w\rangle \ge \underline{\kappa }\langle w,w\rangle \forall w\in X.$$

Lemma 2. 

Assume that $\mathcal{O}$ be an open subset of ${\mathbb{R}}^{p+q}$, Σ be a nonempty compact subset of ${\mathbb{R}}^p$, and $g:\mathcal{O}\to {\mathbb{R}}^q$ be a mapping and $g\in {\mathcal{C}}^{\kappa }$ on $\mathcal{O}$ for some $\kappa \ge 0$. Assume that ${\mathcal{J}}_{w}g(z,w)$ exists, and it is continuous on $\mathcal{O}$. Assume that $\overline{w}\in {\mathbb{R}}^q$ is a vector such that $g(\overline{z},\overline{w})=0$ for $(\overline{z},\overline{w})\in \Sigma$, and the Jacobian ${\mathcal{J}}_{w}g(\overline{z},\overline{w})$ is nonsingular for all $\overline{z}\in \Sigma$. Then, there exist scalars $\epsilon >0$, $\delta >0$ and a mapping $\psi :\mathit{B}(\Sigma ,\epsilon )\to \mathit{B}(\overline{w},\delta )$ such that $\psi \in {\mathcal{C}}^{\kappa }$ on $\mathit{B}(\Sigma ,\epsilon )$, $\overline{w}=\psi \left(\overline{z}\right)$ for all $\overline{z}\in \Sigma$, and $g[z,\psi (z\left)\right]=0$ for all $z\in \mathit{B}(\Sigma ,\epsilon )$. The mapping ψ is unique in the sense that if $g(z,w)=0$, $z\in \mathbf{B}(\Sigma ,\epsilon )$ and $w\in \mathit{B}(\overline{w},\delta )$, then $w=\psi \left(z\right)$. Furthermore, if $\kappa \ge 1$, then we have

$$\mathcal{J}\psi \left(z\right)=-{\mathcal{J}}_{w}g{(z,\psi \left(z\right))}^{-1}{\mathcal{J}}_{z}g(z,\psi \left(z\right)),\forall z\in \mathit{B}(\Sigma ,\epsilon ).$$

2. Properties of the Augmented Lagrangian

Assume that $\overline{x}$ is a given feasible point of Problem (1) and f, h and g are twice differentiable in a neighborhood of $\overline{x}$. The following conditions, which are called Jacobian uniqueness conditions, are needed in our analysis.

Definition 1. 

Jacobian uniqueness conditions at $(\overline{x},\overline{\pi },\overline{Y})\in {\mathbb{R}}^n\times {\mathbb{R}}^q\times {\mathbb{S}}^p$ are the following conditions:

(i) 

The point $(\overline{x},\overline{\pi },\overline{Y})$ satisfies the Karush–Kuhn–Tucker condtions:

$${\nabla }_x\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})=0,h\left(\overline{x}\right)=0,0⪯\overline{Y}\perp g\left(\overline{x}\right)⪯0.$$

(ii) 

The constraint nondegeneracy condition is satisfied at $\overline{x}$:

$$\left(\begin{array}{c}{\mathbb{R}}^q\\ {\mathbb{S}}^p\end{array}\right)=\left(\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\}\\ \mathrm{D}g\left(\overline{x}\right)\end{array}\right){\mathbb{R}}^n+\left(\begin{array}{c}\left\{0\right\}\\ \mathrm{lin}(T_{{\mathbb{S}}_{-}^p}\left(g\left(\overline{x}\right)\right)\end{array}\right),$$

where $\mathrm{lin}\left(K\right)$ denotes the linearity space of a closed convex cone.

(iii) 

The strict complementarity condition at $(\overline{x},\overline{Y})$ holds, namely $\overline{Y}-g\left(\overline{x}\right)\succ 0$.

(iv) 

At $(\overline{x},\overline{\pi },\overline{Y})$, the second-order sufficiency optimality conditions holds, namely for any $d\ne 0$ satisfying $d\in \mathcal{C}\left(\overline{x}\right)$,

$$0<d^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d-2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d\rangle ,$$

where ${\left[g\left(\overline{x}\right)\right]}^{†}$ is the Moore–Penrose pseudoinverse of $g\left(\overline{x}\right)]$ and $\mathcal{C}\left(\overline{x}\right)$ is the critical cone at $\overline{x}$ defined by

$$\mathcal{C}\left(\overline{x}\right)=\{d\in {\mathbb{R}}^n:\nabla f{\left(\overline{x}\right)}^{T}d\le 0,\mathcal{J}h\left(\overline{x}\right)d=0,\mathrm{D}g\left(\overline{x}\right)d\in T_{{\mathbb{S}}_{-}^p}\left(g\left(\overline{x}\right)\right)\}.$$

In this section, we will give some properties of the Jacobian uniqueness conditions of Problem (1) and properties of the augmented Lagrange function under this set of conditions. These properties are crucial for studying the convergence rate of augmented Lagrange method.

Let $(\overline{x},\overline{\pi },\overline{Y})$ be a KKT pair. Assume that (iii) holds; then, $g\left(\overline{x}\right)+\overline{Y}$ is nonsingular. Let the eigenvalues of $g\left(\overline{x}\right)+\overline{Y}$ be ${\lambda }_1,\dots ,{\lambda }_p$ and

$$\gamma =\{i:{\lambda }_i<0\},\alpha =\{i:{\lambda }_i>0\}.$$

Then, an orthogonal matrix $P\in {\mathbb{R}}^p$ exists such that

$$g\left(\overline{x}\right)+\overline{Y}=P\Lambda P^T,\overline{Y}=P\left[\begin{array}{cc}{\Lambda }_{\alpha }& \\ & 0\end{array}\right]P^T=P_{\alpha }{\Lambda }_{\alpha }{P}_{\alpha }^T,g\left(\overline{x}\right)=P\left[\begin{array}{cc}0& \\ & {\Lambda }_{\gamma }\end{array}\right]P^T=P_{\gamma }{\Lambda }_{\gamma }{P}_{\gamma }^T,$$

where

$${\Lambda }_{\alpha }=\mathrm{Diag}({\lambda }_i:i\in \alpha ),{\Lambda }_{\gamma }=\mathrm{Diag}({\lambda }_i:i\in \gamma ),P_{\alpha }=(p_i:i\in \alpha ),P_{\gamma }=(p_i:i\in \gamma ).$$

If Jacobian uniqueness conditions (i)–(iii) hold, then the cone $C\left(\overline{x}\right)$ is reduced to the following subspace

$$C\left(\overline{x}\right)=\{d\in {\mathbb{R}}^n:P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d{P}_{\alpha }=0,\mathcal{J}h\left(\overline{x}\right)d=0\}.$$

If Jacobian uniqueness condition (iv) holds, then there exists ${\beta }_0>0$ such that

$${\beta }_0{\parallel d\parallel }^2\le d^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d-2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d\rangle ,\forall d\in C\left(\overline{x}\right).$$

(4)

In fact, if this is not true, then a sequence $d^j\in C\left(\overline{x}\right)$ with $\parallel d^j\parallel =1$ exists such that

$${\displaystyle \frac{1}{j}=\frac{1}{j}{\parallel d^j\parallel }^2\ge \langle d^j,{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d^j\rangle -2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d^j{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d^j\rangle .}$$

(5)

There exists a subsequence $j_k$ and $d\in {\mathbb{R}}^n$ with $\parallel d\parallel =1$ such that $d^{j_k}\to d$. The closedness of $C\left(\overline{x}\right)$ implies $d\in C\left(\overline{x}\right)$. Taking the limit of (5) along the subsequence $j_k$, we obtain

$$\begin{array}{c}\langle d^{j_k},{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d^{j_k}\rangle -2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d^{j_k}{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d^{j_k}\rangle \hfill \\ \to d^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d-2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d\rangle \le 0,\hfill \end{array}$$

which contradicts Jacobian uniqueness condition (iv).

Define

$$\Phi (x,\pi ,Y)=\left[\begin{array}{c}{\nabla }_x\mathbb{L}(x,\pi ,Y)\\ h\left(x\right)\\ {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+g\left(x\right))-Y\end{array}\right].$$

(6)

Then, the Jacobian of $\Phi (x,\pi ,Y)$, denoted by ${\mathcal{M}}_0(x,\pi ,Y)$, is expressed as

$${\mathcal{M}}_0(x,\pi ,Y)=\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathbb{L}(x,\pi ,Y)& \mathcal{J}h{\left(x\right)}^T& \mathrm{D}g{\left(x\right)}^{*}\\ \mathcal{J}h\left(x\right)& 0& 0\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+g\left(x\right))\mathrm{D}g\left(x\right)& 0& -\mathcal{I}+\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+g\left(x\right))\end{array}\right].$$

(7)

Lemma 3. 

Let $\overline{x}\in {\mathbb{R}}^n$ be a given point and f, h and g be twice continuously differentiable in a neighborhood of $\overline{x}$. Let the Jacobian uniqueness conditions at $(\overline{x},\overline{\pi },\overline{Y})\in {\mathbb{R}}^n\times {\mathbb{R}}^q\times {\mathbb{S}}^p$ are satisfied. Then, ${\mathcal{M}}_0(\overline{x},\overline{\pi },\overline{Y})$ is a nonsingular linear operator.

Proof. 

Consider the equation

$${\mathcal{M}}_0(\overline{x},\overline{\pi },\overline{Y})\left(\begin{array}{c}{d}_x\\ d_{\pi }\\ d_Y\end{array}\right)=0,$$

where $d_x\in {\mathbb{R}}^n,d_{\pi }\in {\mathbb{R}}^q,d_Y\in {\mathbb{S}}^p$. This equation is equivalent to

$$\begin{array}{c}{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x+\mathcal{J}h{\left(\overline{x}\right)}^T{d}_{\pi }+\mathrm{D}g{\left(\overline{x}\right)}^{*}{d}_Y=0,\hfill \\ \mathcal{J}h\left(\overline{x}\right)d_x=0,\hfill \\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Y}+g\left(\overline{x}\right))\mathrm{D}g\left(\overline{x}\right)d_x-d_Y+\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Y}+g\left(\overline{x}\right))d_Y=0.\hfill \end{array}$$

(8)

From the third equality of (8) we have for $H=\mathrm{D}g\left(\overline{x}\right)d_x$ that

$$P\left[\begin{array}{cc}0& {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }\circ P_{\alpha }^T{d}_Y{P}_{\gamma }\\ {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }^T\circ P_{\gamma }^T{d}_Y{P}_{\alpha }& P_{\gamma }^T{d}_Y{P}_{\gamma }\end{array}\right]P^T=P\left[\begin{array}{cc}{P}_{\alpha }^{T}H{P}_{\alpha }& {\mathsf{\Omega }}_{\alpha ,\gamma }\circ P_{\alpha }^{T}H{P}_{\gamma }\\ {\mathsf{\Omega }}_{\alpha ,\gamma }^T\circ P_{\gamma }^{T}H{P}_{\alpha }& 0\end{array}\right]P^T,$$

where

$${\displaystyle {\tilde{\mathsf{\Omega }}}_{ij}=\frac{|{\lambda }_j|}{{\lambda }_i+\left|{\lambda }_j\right|},i\in \alpha ,j\in \gamma ,{\mathsf{\Omega }}_{ij}=\frac{{\lambda }_i}{{\lambda }_i+\left|{\lambda }_j\right|}.}$$

This implies the following relations

$$\begin{array}{c}{P}_{\alpha }^{T}H{P}_{\alpha }=0,P_{\gamma }^T{d}_Y{P}_{\gamma }=0,\hfill \\ {\displaystyle P_{\alpha }^T{d}_Y{P}_{\gamma }={\mathsf{\Theta }}_{\alpha ,\gamma }\circ P_{\alpha }^{T}H{P}_{\gamma },{\mathsf{\Theta }}_{ij}=-\frac{{\lambda }_i}{{\lambda }_j},i\in \alpha ,j\in \gamma .}\hfill \end{array}$$

(9)

From $\mathcal{J}h\left(\overline{x}\right)d_x=0$ and $P_{\alpha }^{T}H{P}_{\alpha }=0$, we have $d_x\in \mathcal{C}\left(\overline{x}\right)$. Multiplying $d_x^T$ to the first equality of (8) we obtain

$$\begin{array}{cc}0\hfill & =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi })d_x+d_x^T\mathcal{J}h{\left(\overline{x}\right)}^T{d}_{\pi }+\langle \mathrm{D}g\left(\overline{x}\right)d_x,d_Y\rangle \hfill \\ & =\langle P^T\mathrm{D}g\left(\overline{x}\right)d_{x}P,P^T{d}_{Y}P\rangle +d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x\hfill \\ & =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x+2\langle P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d_x{P}_{\gamma },P_{\alpha }^T{d}_Y{P}_{\gamma }\rangle \hfill \\ & =2\langle P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d_x{P}_{\gamma },{\mathsf{\Theta }}_{\alpha ,\gamma }\circ P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d_x{P}_{\gamma }\rangle +d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x\hfill \\ & =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x-2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d_x{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d_x\rangle ,\hfill \end{array}$$

which implies $d_x=0$ from Jacobian uniqueness condition (iv). This comes from the fact that $d_x\ne 0$ implies

$$d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x-2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d_x{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d_x\rangle >0$$

from Jacobian uniqueness condition (iv). Then, from the first equality of (8) we obtain

$$\mathcal{J}h{\left(\overline{x}\right)}^T{d}_{\pi }+\mathrm{D}g{\left(\overline{x}\right)}^{*}{d}_Y=0,$$

which is equivalent to

$${\displaystyle \left(\begin{array}{c}{\displaystyle \langle \frac{\partial g}{\partial x_1}\left(\overline{x}\right),d_Y\rangle }\\ ⋮\\ {\displaystyle \langle \frac{\partial g}{\partial x_n}\left(\overline{x}\right),d_Y\rangle }\end{array}\right)+\sum _{j=1}^q{\left[d_{\pi }\right]}_j\nabla h_j\left(\overline{x}\right)=0.}$$

This, from $P_{\gamma }^T{d}_Y{P}_{\gamma }=0$, implies

$${\displaystyle \sum _{j=1}^q{\left[d_{\pi }\right]}_j\nabla h_j\left(\overline{x}\right)+\left(\begin{array}{c}{\displaystyle \langle P_{\alpha }^T\frac{\partial g}{\partial x_1}\left(\overline{x}\right)P_{\alpha },P_{\alpha }^T{d}_Y{P}_{\alpha }\rangle }\\ ⋮\\ {\displaystyle \langle P_{\alpha }^T\frac{\partial g}{\partial x_n}\left(\overline{x}\right)P_{\alpha },P_{\alpha }^T{d}_Y{P}_{\alpha }\rangle }\end{array}\right)+2\left(\begin{array}{c}{\displaystyle \langle P_{\alpha }^T\frac{\partial g}{\partial x_1}\left(\overline{x}\right)P_{\gamma },P_{\alpha }^T{d}_Y{P}_{\gamma }\rangle }\\ ⋮\\ {\displaystyle \langle P_{\alpha }^T\frac{\partial g}{\partial x_n}\left(\overline{x}\right)P_{\gamma },P_{\alpha }^T{d}_Y{P}_{\gamma }\rangle }\end{array}\right)=0.}$$

From Jacobian uniqueness condition (ii) we obtain

$$d_{\pi }=0 \mathrm{and} P_{\alpha }^T{d}_Y{P}_{\alpha \cup \gamma }=0,$$

and this implies $d_{\pi }=0$ and $d_Y=0$. Combining $d_x=0$, we obtain that ${\mathcal{M}}_0(\overline{x},\overline{\pi },\overline{Y})$ is a nonsingular linear operator. □

Proposition 1. 

Let $(\overline{x},\overline{\pi },\overline{Y})$ be a Karush–Kuhn–Tucker point of Problem (1) and the Jacobian uniqueness conditions are satisfied at $(\overline{x},\overline{\pi },\overline{Y})$. Then there exist positive numbers ${\delta }_0>0$ and ${\sigma }_0^{*}>0$ such that ${\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(x,\pi ,Y)$ is positively definite when $\sigma \ge {\sigma }_0^{*}$ and $(x,\pi ,Y)\in {\mathbf{B}}_{{\delta }_0}(\overline{x},\overline{\pi },\overline{Y})$.

Proof. 

It is easy to obtain

$${\nabla }_x{\mathbb{L}}_{\sigma }(x,\pi ,Y)=\nabla f\left(x\right)+\mathrm{D}g{\left(x\right)}^{*}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+\sigma g\left(x\right))+\mathcal{J}h{\left(x\right)}^T(\pi +\sigma h\left(x\right)).$$

If $Y+\sigma g\left(x\right)$ is nonsingular, then ${\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}$ is differentiable at $Y+\sigma g\left(x\right)$ and

$${\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(x,\pi ,Y)={\nabla }_{xx}^2\mathbb{L}(x,\pi +\sigma h\left(x\right),{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+\sigma g\left(x\right)))+\sigma \mathrm{D}g{\left(x\right)}^{*}\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+\sigma g\left(x\right))\mathrm{D}g\left(x\right)+\sigma \mathcal{J}h{\left(x\right)}^T\mathcal{J}h\left(x\right).$$

Define

$${\displaystyle {\overline{\sigma }}_0^{*}=\frac{4}{{\eta }_0^2{\beta }_0}\sum _{i\in \alpha ,j\in \gamma }\sum _{k=1}^n{\left({\displaystyle p_i^T\frac{\partial g\left(\overline{x}\right)}{\partial x_k}{p}_j}\right)}^2,{\eta }_0=min\left\{{\displaystyle \frac{|{\lambda }_j|}{{\lambda }_i}:i\in \alpha ,j\in \gamma }\right\}.}$$

Then, from (4), we obtain for any vector $d\in {\mathbb{R}}^n$,

$$\begin{array}{c}{d}^T{\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(\overline{x},\overline{\pi },\overline{Y})d=d^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y}))d+\sigma \parallel \mathcal{J}h\left(\overline{x}\right){d\parallel }^2+\sigma \langle \mathrm{D}g\left(\overline{x}\right)d,\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Y}+\sigma g\left(\overline{x}\right))\mathrm{D}g\left(\overline{x}\right)d\rangle \hfill \\ =\sigma \parallel \mathcal{J}h\left(\overline{x}\right){d\parallel }^2+\sigma {\parallel P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d{P}_{\alpha }\parallel }^2+\left\{d^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d-2\langle \overline{Y},\mathrm{D}g\left(\overline{x}\right)d{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d\rangle \right\}\hfill \\ {\displaystyle +2\sum _{i\in \alpha ,j\in \gamma }\left({\displaystyle \frac{\sigma {\lambda }_i}{{\lambda }_i+\sigma \left|{\lambda }_j\right|}+\frac{{\lambda }_i}{{\lambda }_j}}\right){\left(p_i^T\mathrm{D}g\left(\overline{x}\right)d{p}_j\right)}^2}\hfill \\ {\displaystyle \ge {\beta }_0{\parallel d\parallel }^2+\sum _{i\in \alpha ,j\in \gamma }\left({\displaystyle \frac{-2{\lambda }_i^2}{({\lambda }_i+\sigma |{\lambda }_j\left|\right)|{\lambda }_j|}}\right){\left(p_i^T\mathrm{D}g\left(\overline{x}\right)d{p}_j\right)}^2+\sigma \parallel \mathcal{J}h\left(\overline{x}\right){d\parallel }^2+\sigma {\parallel P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d{P}_{\alpha }\parallel }^2}\hfill \\ {\displaystyle \ge {\beta }_0{\parallel d\parallel }^2-\frac{2}{{\eta }_0(1+\sigma {\eta }_0)}{\parallel d\parallel }^2\sum _{i\in \alpha ,j\in \gamma }\sum _{k=1}^n{\left({\displaystyle p_i^T\frac{\partial g\left(\overline{x}\right)}{\partial x_k}{p}_j}\right)}^2+\sigma \parallel \mathcal{J}h\left(\overline{x}\right){d\parallel }^2+\sigma {\parallel P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d{P}_{\alpha }\parallel }^2}\hfill \\ {\displaystyle \ge \frac{{\beta }_0}{2}{\parallel d\parallel }^2+\sigma \parallel \mathcal{J}h\left(\overline{x}\right){d\parallel }^2+\sigma {\parallel P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d{P}_{\alpha }\parallel }^2.}\hfill \end{array}$$

This implies for any vector $d\in C\left(\overline{x}\right)$ that

$${\displaystyle d^T{\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(\overline{x},\overline{\pi },\overline{Y})d\ge \frac{{\beta }_0}{2}{\parallel d\parallel }^2.}$$

It follows from (ii) that the linear mapping

$$d\to (\mathcal{J}h\left(\overline{x}\right)d,P_{\alpha }^T\mathrm{D}g\left(\overline{x}\right)d{P}_{\alpha })$$

is onto. Then, we have from Lemma 1 that there exists ${\sigma }_0^{*}>{\overline{\sigma }}_0^{*}$ such that ${\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(\overline{x},\overline{\pi },\overline{Y})$ is positively definite if $\sigma \ge {\sigma }_0^{*}$. Therefore, there exists a positive real number ${\delta }_0>0$ such that ${\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(x,\pi ,Y)$ is positively definite if $(x,\pi ,Y)\in {\mathbf{B}}_{{\delta }_0}(\overline{x},\overline{\pi },\overline{Y})$ and $\sigma \ge {\sigma }_0^{*}$. □

Suppose that $Z(x,Y,t)=g\left(x\right)+(t+1)Y$ is nonsingular such that ${\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}$ is differentiable at $Z(x,Y,t)$. In this case, we define a linear operator:

$$\mathcal{M}(x,\pi ,Y,t)=\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathbb{L}(x,\pi ,Y)& \mathcal{J}h{\left(x\right)}^T& \mathrm{D}g{\left(x\right)}^{*}\\ \mathcal{J}h\left(x\right)& -t{I}_q& 0\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)\mathrm{D}g\left(x\right)& 0& -(t+1)\mathcal{I}+\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)\end{array}\right].$$

(10)

Proposition 2. 

Let $(\overline{x},\overline{\pi },\overline{Y})$ be a Karush–Kuhn–Tucker point of Problem (1) and the Jacobian uniqueness conditions are satisfied at $(\overline{x},\overline{\pi },\overline{Y})$. Then, there exists a positive real number ${\sigma }_1^{*}>{\sigma }_0^{*}$ large enough, such that $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)$ is nonsingular and

$$\parallel \mathcal{M}{(\overline{x},\overline{\pi },\overline{Y},t)}^{-1}\parallel \le {\beta }_0$$

for some positive constant ${\beta }_0>0$ if $t\in [0,t_1^{*}]$, where $t_1^{*}={\left[{\sigma }_1^{*}\right]}^{-1}$.

Proof. 

We divide the proof into two steps.

Step 1:

we prove that for $t>0$ sufficiently small $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)$ is nonsingular.

Since $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},0)={\mathcal{M}}_0(\overline{x},\overline{\pi },\overline{Y})$ we have from Lemma 3 that $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},0)$ is nonsingular. Now, we consider the case where $t>0$. Consider the equation

$$\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)\left(\begin{array}{c}{d}_x\\ d_{\pi }\\ d_Y\end{array}\right)=0,$$

where $d_x\in {\mathbb{R}}^n,d_{\pi }\in {\mathbb{R}}^q,d_Y\in {\mathbb{S}}^p$. This equation is equivalent to

$$\begin{array}{c}{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x+\mathcal{J}h{\left(\overline{x}\right)}^T{d}_{\pi }+\mathrm{D}g{\left(\overline{x}\right)}^{*}{d}_Y=0,\hfill \\ \mathcal{J}h\left(\overline{x}\right)d_x-t{d}_{\pi }=0,\hfill \\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}((1+t)\overline{Y}+g\left(\overline{x}\right))\mathrm{D}g\left(\overline{x}\right)d_x-(1+t)d_Y+\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}((1+t)\overline{Y}+g\left(\overline{x}\right))d_Y=0.\hfill \end{array}$$

(11)

From the second equality of (11), we have

$$d_{\pi }=t^{-1}\mathcal{J}h\left(\overline{x}\right)d_x.$$

From the third equality of (11), we have for $H=\mathrm{D}g\left(\overline{x}\right)d_x$ that

$$P\left[\begin{array}{cc}t{P}_{\alpha }^T{d}_Y{P}_{\alpha }& {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }\left(t\right)\circ P_{\alpha }^T{d}_Y{P}_{\gamma }\\ {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }{\left(t\right)}^T\circ P_{\gamma }^T{d}_Y{P}_{\alpha }& (1+t)P_{\gamma }^T{d}_Y{P}_{\gamma }\end{array}\right]P^T=P\left[\begin{array}{cc}{P}_{\alpha }^{T}H{P}_{\alpha }& {\mathsf{\Omega }}_{\alpha ,\gamma }\left(t\right)\circ P_{\alpha }^{T}H{P}_{\gamma }\\ {\mathsf{\Omega }}_{\alpha ,\gamma }{\left(t\right)}^T\circ P_{\gamma }^{T}H{P}_{\alpha }& 0\end{array}\right]P^T,$$

where

$${\displaystyle {\mathsf{\Omega }}_{ij}\left(t\right)=\frac{(1+t){\lambda }_i}{(1+t){\lambda }_i+\left|{\lambda }_j\right|},{\tilde{\mathsf{\Omega }}}_{ij}\left(t\right)=\frac{(1+t)[t{\lambda }_i+|{\lambda }_j\left|\right]}{(1+t){\lambda }_i+\left|{\lambda }_j\right|},i\in \alpha ,j\in \gamma .}$$

This implies the following relations

$$\begin{array}{c}{P}_{\alpha }^T{d}_Y{P}_{\alpha }=t^{-1}{P}_{\alpha }^{T}H{P}_{\alpha },P_{\gamma }^T{d}_Y{P}_{\gamma }=0,\hfill \\ {\displaystyle P_{\alpha }^T{d}_Y{P}_{\gamma }={\mathsf{\Theta }}_{\alpha ,\gamma }\left(t\right)\circ P_{\alpha }^{T}H{P}_{\gamma },{\mathsf{\Theta }}_{ij}\left(t\right)=\frac{{\lambda }_i}{t{\lambda }_i+\left|{\lambda }_j\right|},i\in \alpha ,j\in \gamma .}\hfill \end{array}$$

(12)

Then, multiplying $d_x^T$ to the first equality of (11), we obtain

$$\begin{array}{cc}0\hfill & =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x+d_x^T\mathcal{J}h{\left(\overline{x}\right)}^T{d}_{\pi }+\langle \mathrm{D}g\left(\overline{x}\right)d_x,d_x\rangle \hfill \\ & {\displaystyle =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x+t^{-1}\parallel \mathcal{J}h\left(\overline{x}\right)d_x{\parallel }^2+t^{-1}{\parallel P_{\alpha }^{T}H{P}_{\alpha }\parallel }^2+2\sum _{i\in \alpha ,j\in \gamma }{\mathsf{\Theta }}_{ij}\left(t\right)H_{ij}^2}\hfill \\ & {\displaystyle =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x-2\sum _{i\in \alpha ,j\in \gamma }\frac{{\lambda }_i}{{\lambda }_j}{H}_{ij}^2+t^{-1}{\parallel \mathcal{J}h\left(\overline{x}\right)d_x\parallel }^2}\hfill \\ & {\displaystyle +2\sum _{i\in \alpha ,j\in \gamma }\left[{\mathsf{\Theta }}_{ij}\left(t\right)+\frac{{\lambda }_i}{{\lambda }_j}\right]H_{ij}^2+t^{-1}{\parallel P_{\alpha }^{T}H{P}_{\alpha }\parallel }^2}\hfill \\ & =d_x^T{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})d_x-2\langle Y^{*},\mathrm{D}g\left(\overline{x}\right)d_x{\left[g\left(\overline{x}\right)\right]}^{†}\mathrm{D}g\left(\overline{x}\right)d_x\rangle \hfill \\ & {\displaystyle +t^{-1}\parallel \mathcal{J}h\left(\overline{x}\right)d_x{\parallel }^2+2\sum _{i\in \alpha ,j\in \gamma }\left[\frac{t{\lambda }_i^2}{t{\lambda }_i{\lambda }_j-{\lambda }_j^2}\right]H_{ij}^2+t^{-1}{\parallel P_{\alpha }^{T}H{P}_{\alpha }\parallel }^2}\hfill \\ & =d_x^T{\nabla }_{xx}^2{L}_{t^{-1}}(\overline{x},\overline{\pi },\overline{Y})d_x.\hfill \end{array}$$

which implies $d_x=0$ from Proposition 1 when $t<{\left[{\sigma }_0^{*}\right]}^{-1}$. Therefore, we obtain $d_x=0$, $d_{\pi }=0$, and $d_Y=0$ so that $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)$ is nonsingular when $t<{\left[{\sigma }_0^{*}\right]}^{-1}$.

Step 2:

We prove that

$$\parallel \mathcal{M}{(\overline{x},\overline{\pi },\overline{Y},t)}^{-1}\parallel \le {\beta }_0$$

for some positive constant ${\beta }_0>0$ if $t\ge 0$ small enough.

Noting, for $\overline{Z}=\overline{Y}+g\left(\overline{x}\right)$ we have $Z(\overline{x},\overline{Y},t)=t\overline{Y}+\overline{Z}$ and $Z(\overline{x},\overline{Y},0)=\overline{Z}$. Therefore, we get

$$\begin{array}{c}\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},0)\hfill \\ =\left[\begin{array}{ccc}0& 0& 0\\ 0& -t{I}_q& 0\\ [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Z}+t\overline{Y})-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(\overline{Z}\right)]\mathrm{D}g\left(\overline{x}\right)& 0& -t\mathcal{I}+[\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Z}+t\overline{Y})-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(\overline{Z}\right)].\end{array}\right]\hfill \end{array}$$

(13)

For any $(d_x,d_{\pi },d_Y)\in {\mathbb{R}}^n\times {\mathbb{R}}^q\times {\mathbb{S}}^p$ we have that

$$[\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},0)]\left(\begin{array}{c}{d}_x\\ d_{\pi }\\ d_Y\end{array}\right)=\left[\begin{array}{c}0\\ -t{d}_{\pi }\\ [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Z}+t\overline{Y})-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(\overline{Z}\right)][\mathrm{D}g\left(\overline{x}\right)d_x+d_Y]-t{d}_Y\end{array}\right].$$

(14)

For any matrix $H\in {\mathbb{S}}^p$, we have

$$[\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Z}+t\overline{Y})-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(\overline{Z}\right)]H=P\left[\begin{array}{cc}0& {\overline{\mathsf{\Omega }}}_{\alpha ,\gamma }\left(t\right)\circ P_{\alpha }^{T}H{P}_{\gamma }\\ {\overline{\mathsf{\Omega }}}_{\alpha ,\gamma }{\left(t\right)}^T\circ P_{\gamma }^{T}H{P}_{\alpha }& 0\end{array}\right]P^T,$$

where

$${\displaystyle {\overline{\mathsf{\Omega }}}_{ij}\left(t\right)=\frac{t{\lambda }_i\left|{\lambda }_j\right|}{({\lambda }_i+|{\lambda }_j\left|\right)((t+1){\lambda }_i+|{\lambda }_j\left|\right)}\le \frac{t}{4},i\in \alpha ,j\in \gamma .}$$

(15)

We have from (14) and (15) that

$$\begin{array}{c}{∥[\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},0)]\left(\begin{array}{c}{d}_x\\ d_{\pi }\\ d_Y\end{array}\right)∥}^2\hfill \\ =t^2\parallel d_{\pi }{\parallel }^2+\parallel [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Z}+t\overline{Y})-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(\overline{Z}\right)]\mathrm{D}g\left(\overline{x}\right)d_x{\parallel }^2+{\parallel [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\overline{Z}+t\overline{Y})-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(\overline{Z}\right)]d_Y-t{d}_Y\parallel }^2\hfill \\ {\displaystyle \le t^2\parallel d_{\pi }{\parallel }^2+\frac{t^2}{4}\parallel \mathrm{D}g\left(\overline{x}\right){\parallel }^2\parallel d_x{\parallel }^2+t^2{\parallel d_Y\parallel }^2}\hfill \\ \le max\left\{{\displaystyle 1,\frac{1}{4}{\parallel \mathrm{D}g\left(\overline{x}\right)\parallel }^2}\right\}{t}^2{∥\left(\begin{array}{c}{d}_x\\ d_{\pi }\\ d_Y\end{array}\right)∥}^2.\hfill \end{array}$$

Thus, we have, for $t\in [0,{\left[c_0^{*}\right]}^{-1})$, that

$$\parallel [\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},0)\parallel \le max\left\{{\displaystyle 1,\frac{1}{2}\parallel \mathrm{D}g\left(\overline{x}\right)\parallel }\right\}t.$$

Therefore, there exists a sufficiently large positive number ${\sigma }_1^{*}>{\sigma }_0^{*}$, for $t_1^{*}={\left[{\sigma }_1^{*}\right]}^{-1}$, if $t\in [0,t_1^{*}]$; then, $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)$ is nonsingular and

$$\parallel \mathcal{M}{(\overline{x},\overline{\pi },\overline{Y},t)}^{-1}\parallel \le {\beta }_0$$

for some positive constant ${\beta }_0>0$. The proof is complete. □

Proposition 3. 

The corresponding Löwner operator F is twice (continuously) differentiable at X if and only if f is twice (continuously) differentiable at ${\lambda }_i\left(X\right)$, $i=1,\dots ,p$.

Proposition 4. 

Let $(\overline{x},\overline{\pi },\overline{Y})$ be a Karush–Kuhn–Tucker point of Problem (1) and the Jacobian uniqueness conditions are satisfied at $(\overline{x},\overline{\pi },\overline{Y})$. Then, there exist ${\beta }_1\ge {\beta }_0$, ${\delta }_1\in (0,{\delta }_0)$, and ${\sigma }_2^{*}\ge {\sigma }_1^{*}$, for $t_2^{*}={\left[{\sigma }_2^{*}\right]}^{-1}$, $\mathcal{M}(x,\pi ,Y,t)$ is nonsingular and

$${\parallel \mathcal{M}(x,\pi ,Y,t)}^{-1}\parallel \le {\beta }_1$$

if $(x,\pi ,Y)\in {\mathit{B}}_{{\delta }_1}(\overline{x},\overline{\pi },\overline{Y})$ and $t\in [0,t_2^{*})$.

Proof. 

We have from Proposition 2 that the operator $\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t_1^{*})$ is nonsingular. Since the norm of $\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}$ is less than 1 and

$$\begin{array}{c}[\mathcal{M}(x,\pi ,Y,t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)]\hfill \\ =\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathbb{L}(x,\pi ,Y)-{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})& \mathcal{J}h{\left(x\right)}^T-\mathcal{J}h{\left(\overline{x}\right)}^T& \mathrm{D}g{\left(x\right)}^{*}-\mathrm{D}g{\left(\overline{x}\right)}^{*}\\ \mathcal{J}h\left(x\right)-\mathcal{J}h\left(\overline{x}\right)& 0& 0\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)\mathrm{D}g\left(x\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Z(\overline{x},\overline{Y},t)\mathrm{D}g\left(\overline{x}\right)& 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Z(\overline{x},\overline{Y},t)\end{array}\right],\hfill \end{array}$$

we have

$$\begin{array}{c}\parallel \mathcal{M}(x,\pi ,Y,t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)\parallel \hfill \\ \le \parallel {\nabla }_{xx}^2\mathbb{L}(x,\pi ,Y)-{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})\parallel +2\parallel \mathcal{J}h\left(x\right)-\mathcal{J}h\left(\overline{x}\right)\parallel +\parallel \mathrm{D}g{\left(x\right)}^{*}-\mathrm{D}g{\left(\overline{x}\right)}^{*}\parallel \hfill \\ +\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Z(\overline{x},\overline{Y},t)\parallel +\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)\mathrm{D}g\left(x\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)\mathrm{D}g\left(\overline{x}\right)\parallel \hfill \\ \le \parallel {\nabla }_{xx}^2\mathbb{L}(x,\pi ,Y)-{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})\parallel +2\parallel \mathcal{J}h\left(x\right)-\mathcal{J}h\left(\overline{x}\right)\parallel +\parallel \mathrm{D}g{\left(x\right)}^{*}-\mathrm{D}g{\left(\overline{x}\right)}^{*}\parallel \hfill \\ +\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)[\mathrm{D}g\left(x\right)-\mathrm{D}g\left(\overline{x}\right)]+[\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)]\mathrm{D}g\left(\overline{x}\right)\parallel \hfill \\ +\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Z(\overline{x},\overline{Y},t)\parallel \hfill \\ \le \parallel {\nabla }_{xx}^2\mathbb{L}(x,\pi ,Y)-{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})\parallel +2\parallel \mathcal{J}h\left(x\right)-\mathcal{J}h\left(\overline{x}\right)\parallel \hfill \\ +2\parallel [\mathrm{D}g\left(x\right)-\mathrm{D}g\left(\overline{x}\right)]\parallel +\parallel [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)]\mathrm{D}g\left(\overline{x}\right)\parallel \hfill \\ +\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(x,Y,t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Z(\overline{x},\overline{Y},t)\parallel .\hfill \end{array}$$

Since ${\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}$ is twice continuously differentiable at $Z(\overline{x},\overline{Y},t)$ we obtain

$${\displaystyle \underset{(x,\pi ,Y)\to (\overline{x},\overline{\pi },\overline{Y})}{lim}\mathcal{M}(x,\pi ,Y,t)=\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)}$$

(16)

for $t\in [0,t_0^{*})$. For $H\in {\mathbb{S}}^p$ we obtain

$$[\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t_1^{*})\right]H=P\left[\begin{array}{cc}0& {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }\circ P_{\alpha }^{T}H{P}_{\gamma }\\ {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }^T\circ P_{\alpha }^{T}H{P}_{\gamma }^T& 0\end{array}\right]P^T,$$

where

$${\displaystyle {\tilde{\mathsf{\Omega }}}_{\alpha ,\gamma }=({\tilde{\mathsf{\Omega }}}_{ij}:i\in \alpha ,j\in \gamma ),{\tilde{\mathsf{\Omega }}}_{ij}=\frac{(1+t){\lambda }_i}{|{\lambda }_j|+(1+t){\lambda }_i}-\frac{(1+t_1^{*}){\lambda }_i}{|{\lambda }_j|+(1+t_1^{*}){\lambda }_i}.}$$

Note for $t\in [0,t_1^{*})$ that

$$\begin{array}{cc}|{\tilde{\mathsf{\Omega }}}_{ij}|\hfill & {\displaystyle =\frac{|t-t_1^{*}|{\lambda }_i\left|{\lambda }_j\right|}{\left(\right|{\lambda }_j|+(1+t){\lambda }_i\left)\right(|{\lambda }_j|+(1+t_1^{*}){\lambda }_i)}}\hfill \\ & {\displaystyle <\frac{|t-t_1^{*}|{\lambda }_i\left|{\lambda }_j\right|}{\left(\right|{\lambda }_j{|+{\lambda }_i)}^2}}\hfill \\ & {\displaystyle \le \frac{t_1^{*}-t}{4},}\hfill \end{array}$$

we have

$$\parallel [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t_1^{*})\right]\parallel \le (t_1^{*}-t))/2.$$

Therefore, from

$$\begin{array}{c}\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)]-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t_1^{*})\hfill \\ =\left[\begin{array}{ccc}0& 0& 0\\ 0& [t_1^{*}-t]I_q& 0\\ [\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t_1^{*})\right]\mathrm{D}g\left(\overline{x}\right)& 0& [t_1^{*}-t]\mathcal{I}+[\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t)\right)-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z(\overline{x},\overline{Y},t_1^{*})\right]\end{array}\right]\hfill \end{array}$$

we get

$$\parallel \mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)]-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t_0)\parallel \le {\alpha }_0[t_0-t].$$

(17)

where

$${\alpha }_0=max\{3/2,\parallel \mathrm{D}g\left(\overline{x}\right)\parallel /2\}.$$

As $\mathcal{M}(x,\pi ,Y,t)$ can be expressed as

$$\mathcal{M}(x,\pi ,Y,t)=\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t_1^{*})+[\mathcal{M}(x,\pi ,Y,t)-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)]+\left[\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)\right]-\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},,t_1^{*})],$$

we have from (16) and (17) that there exist ${\delta }_1\in (0,{\delta }_0)$ and ${\beta }_1>0$ and for sufficiently small $t_2^{*}\in (0,t_1^{*}]$, $\mathcal{M}(x,\pi ,Y,t)$ is nonsingular if $\forall (x,\pi ,Y)\in {\mathbf{B}}_{{\delta }_1}(\overline{x},\overline{\pi },\overline{Y}),\forall t\in [0,t_2^{*})$, and ${\parallel \mathcal{M}(x,\pi ,Y,t)}^{-1}\parallel \le {\beta }_1$ if $(x,\pi ,Y)\in {\mathbf{B}}_{{\delta }_1}(\overline{x},\overline{\pi },\overline{Y}),\forall t\in [0,t_2^{*})$. □

3. The Convergence Rate of the Augmented Lagrange Method

In this section, we focus on the local convergence of the augmented Lagrange method for nonlinear semidefinite optimization problems under the Jacobian uniqueness conditions. Now we estimate the solution error of the augmented Lagrange subproblem

$$\underset{x}{min}{\mathbb{L}}_{\sigma }(x,\pi ,Y)$$

and the error for the updating multiplier when $(\pi ,Y)$ is around $(\overline{\pi },\overline{Y})$. The local convergence and the linear rate of multipliers can be obtained by using these estimates.

For a real number $s>0$ define

$$D({\sigma }^{*},s)=\left\{(\pi ,Y,\sigma ):\parallel (\pi ,Y)-(\overline{\pi },\overline{Y})\parallel \le s\sigma ,\sigma \ge {\sigma }^{*}\right\}.$$

Theorem 1. 

Let Jacobian uniqueness conditions be satisfied at $(\overline{\pi },\overline{Y})$. Then, there exist $ϵ>0$ and $\beta >0$ and $\delta >0$ such that for any $(\pi ,Y,\sigma )\in D({\sigma }^{*},\delta )$ the problem

$$\begin{array}{cc}min\hfill & {\mathbb{L}}_{\sigma }(x,\pi ,Y)\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & x\in {\mathit{B}}_{ϵ}\left(\overline{x}\right)\hfill \end{array}$$

(18)

has a unique solution $x(\pi ,Y,\sigma )$, which is smooth on $\mathrm{int}D({\sigma }^{*},\delta )$. Moreover, for $\forall (\pi ,Y,\sigma )\in \mathrm{int}D({\sigma }^{*},\delta )$,

$$\begin{array}{c}{\displaystyle \parallel x(\pi ,Y,\sigma )-\overline{x}\parallel \le \frac{\beta }{\sigma }\parallel (\pi ,Y)-(\overline{\pi },\overline{Y})\parallel ,}\hfill \\ {\displaystyle \parallel \tilde{\pi }(\pi ,Y,\sigma )-\overline{\pi }\parallel \le \frac{\beta }{\sigma }\parallel (\pi ,Y)-(\overline{\pi },\overline{Y})\parallel ,}\hfill \\ {\displaystyle \parallel \tilde{Y}(\pi ,Y,\sigma )-\overline{Y}\parallel \le \frac{\beta }{\sigma }\parallel (\pi ,Y)-(\overline{\pi },\overline{Y})\parallel .}\hfill \end{array}$$

(19)

where

$$\tilde{\pi }(\pi ,Y,\sigma )=\pi +\sigma h\left(x(\pi ,Y,\sigma )\right),\tilde{Y}(\pi ,Y,\sigma )={\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+\sigma g\left(x(\pi ,Y,\sigma )\right)).$$

Proof. 

If x is a local minimum point of Problem

$${\mathbb{L}}_{\sigma }(·,\pi ,Y),$$

then, from the definition of $(\tilde{\pi },\tilde{Y})$, we obtain

$$\begin{array}{c}\nabla f\left(x\right)+\mathcal{J}h{\left(x\right)}^T\tilde{\pi }+\mathrm{D}g{\left(x\right)}^{*}\tilde{Y}=0,\hfill \\ {\displaystyle h\left(x\right)+\frac{1}{\sigma }(\pi -\tilde{\pi })=0,}\hfill \\ {\displaystyle {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y}\right)-\frac{1}{\sigma }\tilde{Y}=0.}\hfill \end{array}$$

(20)

Define ${\displaystyle \zeta =\frac{1}{\sigma }[\pi -\overline{\pi }]}$, ${\displaystyle U=\frac{1}{\sigma }[Y-\overline{Y}]}$ and ${\displaystyle t=\frac{1}{\sigma }}$, note

$$\begin{array}{cc}{\displaystyle {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y}\right)=\frac{1}{\sigma }\tilde{Y}}\hfill & {\displaystyle ⟺{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\displaystyle \frac{1}{\sigma }\tilde{Y}+\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y-\frac{1}{\sigma }\tilde{Y}}\right)}\right)=\frac{1}{\sigma }\tilde{Y}}\hfill \\ & {\displaystyle ⟺\frac{1}{\sigma }\tilde{Y}\in N_{{\mathbb{S}}_{-}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y-\frac{1}{\sigma }\tilde{Y}}\right)}\hfill \\ & ⟺\left({\displaystyle 1+\frac{1}{\sigma }}\right)\tilde{Y}\in N_{{\mathbb{S}}_{-}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y-\frac{1}{\sigma }\tilde{Y}}\right)\hfill \\ & {\displaystyle ⟺g\left(x\right)+\frac{1}{\sigma }Y-\frac{1}{\sigma }\tilde{Y}={\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y+\tilde{Y}}\right)}\hfill \\ & =\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y+\tilde{Y}}\right)-{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y+\tilde{Y}}\right)\hfill \\ & {\displaystyle ⟺\tilde{Y}+\frac{1}{\sigma }\tilde{Y}={\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\displaystyle g\left(x\right)+\frac{1}{\sigma }Y+\tilde{Y}}\right),}\hfill \end{array}$$

then, the system (20) is equivalent to $\mathbb{F}(x,\tilde{\pi },\tilde{Y};\zeta ,U,t)=0$, where

$$\mathbb{F}(x,\tilde{\pi },\tilde{Y};\zeta ,U,t)=\left[\begin{array}{c}\nabla f\left(x\right)+\mathcal{J}h{\left(x\right)}^T\tilde{\pi }+\mathrm{D}g{\left(x\right)}^{*}\tilde{Y}\hfill \\ h\left(x\right)+\zeta +t\overline{\pi }-t\tilde{\pi }\hfill \\ {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(g\left(x\right)+\tilde{Y}+U+t\overline{Y}\right)-(1+t)\tilde{Y}\hfill \end{array}\right].$$

Obviously, we have

$$\mathbb{F}(\overline{x},\overline{\pi },\overline{Y};0,0,t)=0,\forall t\in [0,t_2^{*}/2],$$

and

$${\mathrm{D}}_{(x,\tilde{\pi },\tilde{Y})}\mathbb{F}(\overline{x},\overline{\pi },\overline{Y};0,0,t)=\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathbb{L}(\overline{x},\overline{\pi },\overline{Y})& \mathcal{J}h{\left(\overline{x}\right)}^T& \mathrm{D}g{\left(\overline{x}\right)}^{*}\\ \mathcal{J}h\left(\overline{x}\right)& -tI& 0\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(g\left(\overline{x}\right)+(1+t)\overline{Y}\right)\mathrm{D}g\left(\overline{x}\right)& 0& -(1+t)\mathcal{I}+\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(g\left(\overline{x}\right)+(1+t)\overline{Y}\right)\end{array}\right].$$

Obviously, from the definition $\mathcal{M}(x,\pi ,Y,t)$ in (10), we have ${\mathrm{D}}_{(x,\tilde{\pi },\tilde{Y})}\mathbb{F}(\overline{x},\overline{\pi },\overline{Y};0,0,t)=\mathcal{M}(\overline{x},\overline{\pi },\overline{Y},t)$. Then, from Proposition 4, we have that ${\mathrm{D}}_{(x,\tilde{\pi },\tilde{Y})}\mathbb{F}(\overline{x},\overline{\pi },\overline{Y};0,0,t)$ is nonsingular when $t\in [0,t_2^{*})$.

Define $t^{*}=t_2^{*}/2$ and ${\sigma }^{*}={\left[t^{*}\right]}^{-1}$ and

$$K=\left\{0\right\}\times \left\{0\right\}\times [0,t^{*}]\subset {\mathbb{R}}^n\times {\mathbb{S}}^p\times \mathbb{R}.$$

From the implicit function theorem, we have that there exists $\delta \in (0,t_2^{*}/2)$ with $\delta <{\delta }_1$, $0<ϵ<{\delta }_1$ and mapping

$$(\widehat{x}(·),\widehat{\pi }(·),\widehat{Y}(·)):\mathbf{B}(K,\delta )\to {\mathbf{B}}_{ϵ}\left((\overline{x},\overline{\pi },\overline{Y})\right),$$

which is smooth on $\mathrm{int}\mathbf{B}(K,\delta )$ and satisfies

$$\begin{array}{c}(\overline{x},\overline{\pi },\overline{\lambda })=(\widehat{x}(0,0,t),\widehat{\pi }(0,0,t),\widehat{Y}(0,0,t)),\hfill \\ \mathbb{F}(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t);\zeta ,U,t)=0\forall (\zeta ,U,t)\in \mathbf{B}(K,\delta ).\hfill \end{array}$$

(21)

From Propositions 1 and 4, we may choose $\delta >0$ and $ϵ>0$ small enough such that constraint nondegeneracy condition holds at $\widehat{x}(\zeta ,U,t)$, ${\nabla }_{xx}^2{\mathbb{L}}_{t^{-1}}(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t)$, $\widehat{Y}(\zeta ,U,t))$ is positively definite and $\parallel \mathcal{M}{(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t),t)}^{-1}\parallel \le {\beta }_1$ for all $(\zeta ,U,t)\in \mathbf{B}(K,\delta )$.

Differentiating the three equations in (21) with respect to $(\zeta ,U,t)$, we obtain

$$\begin{array}{c}\mathcal{M}(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t),t){\mathrm{D}}_{(\zeta ,U,t)}\left[\begin{array}{c}\widehat{x}(\zeta ,U,t)\\ \widehat{\pi }(\zeta ,U,t)\\ \widehat{Y}(\zeta ,U,t)\end{array}\right]\hfill \\ =\left[\begin{array}{ccc}0& 0& 0\\ -I_q& 0& \widehat{\pi }(\zeta ,U,t)-\overline{\pi }\\ 0& -\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\widehat{Z}}_t(\zeta ,U,t)\right)& \widehat{Y}-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left({\widehat{Z}}_t(\zeta ,U,t)\right)\overline{Y}\end{array}\right],\hfill \end{array}$$

(22)

where ${\widehat{Z}}_t(\zeta ,U,t)=g\left(\widehat{x}(\zeta ,U,t)\right)+\widehat{Y}(\zeta ,U,t)+U+t\overline{Y}$. Define $z\left(s\right)=(s\zeta ,sU,st)$ and $\widehat{Z}\left(s\right)=g\left(\widehat{x}\left(z\left(s\right)\right)\right)+\widehat{Y}\left(z\left(s\right)\right)+U+s\overline{Y}$. Then, we have from (22), for $t\in [0,t_2^{*}/2]$ that

$$\begin{array}{c}{\displaystyle \left[\begin{array}{c}\widehat{x}(\zeta ,Y,t)-\overline{x}\\ \widehat{\pi }(\zeta ,Y,t)-\overline{\pi }\\ \widehat{Y}(\zeta ,Y,t)-\overline{Y}\end{array}\right]=\left[\begin{array}{c}\widehat{x}(\zeta ,Y,t)-\widehat{x}(0,0,0)\\ \widehat{\pi }(\zeta ,Y,t)-\widehat{\pi }(0,0,0)\\ \widehat{Y}(\zeta ,Y,t)-\widehat{Y}(0,0,0)\end{array}\right]={\int }_0^1\mathcal{M}{(\widehat{x}\left(z\left(s\right)\right),\widehat{\pi }\left(z\left(s\right)\right),\widehat{Y}\left(z\left(s\right)\right),s)}^{-1}\times }\hfill \\ \times \left[\begin{array}{ccc}0& 0& 0\\ -I& 0& \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }\\ 0& -\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z\left(s\right)\right)& \widehat{Y}-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z\left(s\right)\right)\overline{Y}\end{array}\right]\left[\begin{array}{c}\zeta \\ U\\ t\end{array}\right]\mathrm{d}s\hfill \\ {\displaystyle ={\int }_0^1\mathcal{M}{(\widehat{x}\left(z\left(s\right)\right),\widehat{\pi }\left(z\left(s\right)\right),\widehat{Y}\left(z\left(s\right)\right),s)}^{-1}\left[\begin{array}{c}0\\ -\zeta +(\widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi })t\\ -\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z\left(s\right)\right)U+(\widehat{Y}\left(z\left(s\right)\right)-\overline{Y})t\end{array}\right]\mathrm{d}s}\hfill \\ {\displaystyle +{\int }_0^1\mathcal{M}{(\widehat{x}\left(z\left(s\right)\right),\widehat{\pi }\left(z\left(s\right)\right),\widehat{Y}\left(z\left(s\right)\right),s)}^{-1}\left[\begin{array}{c}0\\ 0\\ t\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}\left(Z\left(s\right)\right)\overline{Y}\end{array}\right]\mathrm{d}s}\hfill \end{array}$$

(23)

Noting that $\parallel \mathcal{M}{(\widehat{x}\left(z\left(s\right)\right),\widehat{\pi }\left(z\left(s\right)\right),\widehat{Y}\left(z\left(s\right)\right),s)}^{-1}\parallel \le {\beta }_1$ for $(\zeta ,U,t)\in \mathbf{B}(K,\delta )$ and $s\in [0,1]$, we obtain from (23) and $\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(g(\widehat{x}\left(z\left(s\right)\right)+\widehat{Y}\left(z\left(s\right)\right)+U+s\overline{Y})\parallel \le 1$ that

$$\begin{array}{c}\parallel \widehat{x}(\zeta ,Y,t)-\overline{x}{\parallel }^2+\parallel \widehat{\pi }(\zeta ,Y,t)-\overline{\pi }{\parallel }^2+{\parallel \widehat{Y}(\zeta ,Y,t)-\overline{Y}\parallel }^2\hfill \\ {\displaystyle \le 4{\beta }_1^2{\int }_0^1\left[{\parallel \zeta \parallel }^2+\parallel \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }{\parallel }^2{t}^2+{\parallel U\parallel }^2+{\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}\parallel }^2{t}^2\right]\mathrm{d}s}\hfill \\ {\displaystyle +2{\beta }_1^2{\int }_0^1\left[\parallel \left(\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\widehat{x}\left(z\left(s\right)\right)\right)+\widehat{Y}\left(z\left(s\right)\right)+U+s\overline{Y})\overline{Y}\right){\parallel }^2{t}^2\right]\mathrm{d}s.}\hfill \end{array}$$

(24)

Noting that ${\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}$ is twice continuously differentiable at $g\left(\overline{x}\right)+(s+1)\overline{Y}$, we have

$$\begin{array}{c}\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\widehat{x}\left(z\left(s\right)\right)\right)+\widehat{Y}\left(z\left(s\right)\right)+U+s\overline{Y})\overline{Y}\hfill \\ =\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\overline{x}\right)+(s+1)\overline{Y})\overline{Y}+{\mathrm{D}}^2{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\overline{x}\right)+(s+1)\overline{Y})[\overline{Y},g\left(\widehat{x}\left(z\left(s\right)\right)\right)+U-g\left(\overline{x}\right)+\widehat{Y}\left(z\left(s\right)\right)-\overline{Y}]\hfill \\ +\mathrm{o}(\parallel g\left(\widehat{x}\left(z\left(s\right)\right)\right)+U-g\left(\overline{x}\right)+\widehat{Y}\left(z\left(s\right)\right)-\overline{Y}\parallel ).\hfill \end{array}$$

It is easy to check the equality $\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\overline{x}\right)+(s+1)\overline{Y})\overline{Y}=0$. Then, when $\delta >0$ is chosen small enough, there exists a positive constant ${\kappa }_0>0$ such that

$$\begin{array}{c}\parallel \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\widehat{x}\left(z\left(s\right)\right)\right)+\widehat{Y}\left(z\left(s\right)\right)+U+s\overline{Y})\overline{Y}\parallel \hfill \\ \le 2\parallel {\mathrm{D}}^2{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}(g\left(\overline{x}\right)+(s+1)\overline{Y})[\overline{Y},g\left(\widehat{x}\left(z\left(s\right)\right)\right)+U-g\left(\overline{x}\right)+\widehat{Y}\left(z\left(s\right)\right)-\overline{Y}]\parallel \hfill \\ \le {\kappa }_0[\parallel U\parallel +\parallel \widehat{x}\left(z\left(s\right)\right)-\overline{x}\parallel +\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}\parallel ]\hfill \end{array}$$

when $(\zeta ,U,t)\in \mathbf{B}(K,\delta )$ and $s\in [0,1]$.

Combining this estimate with (24), we obtain

$$\begin{array}{c}\parallel \widehat{x}(\zeta ,Y,t)-\overline{x}{\parallel }^2+\parallel \widehat{\pi }(\zeta ,Y,t)-\overline{\pi }{\parallel }^2+{\parallel \widehat{Y}(\zeta ,Y,t)-\overline{Y}\parallel }^2\hfill \\ \le 4{\beta }_1^2{\parallel \zeta \parallel }^2+4{\beta }_1^2[1+{\kappa }_0^2{t}^2]{\parallel U\parallel }^2+\hfill \\ {\displaystyle +(1+2{\kappa }_0^2){\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}\parallel }^2+4{\beta }_1^2{t}^2{\int }_0^1\left[\parallel \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }{\parallel }^2+2{\kappa }_0^2{\parallel \widehat{x}\left(z\left(s\right)\right)-\overline{x}\parallel }^2\right]\mathrm{d}s.}\hfill \end{array}$$

(25)

Substituting $(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t)),\widehat{Y}(\zeta ,U,t))$ by $(\widehat{x}\left(z\left(s\right)\right),\widehat{\pi }\left(z\left(s\right)\right),\widehat{Y}\left(z\left(s\right)\right))$ in (25) yields

$$\begin{array}{c}\parallel \widehat{x}\left(z\left(s\right)\right)-\overline{x}{\parallel }^2+\parallel \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }{\parallel }^2+{\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}\parallel }^2\hfill \\ \le 4{\beta }_1^2{\parallel \zeta \parallel }^2+4{\beta }_1^2[1+{\kappa }_0^2{t}^2]{\parallel U\parallel }^2+\hfill \\ {\displaystyle +(1+2{\kappa }_0^2){\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}\parallel }^2+4{\beta }_1^2{t}^2{\int }_0^1\left[\parallel \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }{\parallel }^2+2{\kappa }_0^2{\parallel \widehat{x}\left(z\left(s\right)\right)-\overline{x}\parallel }^2\right]\mathrm{d}s.}\hfill \end{array}$$

(26)

Since the choice of $s\in [0,1]$ in (26) is arbitrary, we obtain

$$\begin{array}{c}{\displaystyle \underset{0\le s\le 1}{max}\left\{\parallel \widehat{x}\left(z\left(s\right)\right)-\overline{x}{\parallel }^2+\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}{\parallel }^2+{\parallel \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }\parallel }^2\right\}}\hfill \\ \le 4{\beta }_1^2{\parallel \zeta \parallel }^2+4{\beta }_1^2[1+{\kappa }_0^2{t}^2]{\parallel U\parallel }^2\hfill \\ {\displaystyle +4{\beta }_1^2(1+2{\kappa }_0^2)t^2·\underset{0\le s\le 1}{max}\left\{\parallel \widehat{x}\left(z\left(s\right)\right)-\overline{x}{\parallel }^2+\parallel \widehat{Y}\left(z\left(s\right)\right)-\overline{Y}{\parallel }^2+{\parallel \widehat{\pi }\left(z\left(s\right)\right)-\overline{\pi }\parallel }^2\right\},}\hfill \end{array}$$

which implies

$$\begin{array}{c}\parallel \widehat{x}(\zeta ,Y,t)-\overline{x}{\parallel }^2+\parallel \widehat{\pi }(\zeta ,Y,t)-\overline{\pi }{\parallel }^2+{\parallel \widehat{Y}(\zeta ,Y,t)-\overline{Y}\parallel }^2\hfill \\ {\displaystyle \le \frac{4{\beta }_1^2[1+{\kappa }_0^2{t}^2]}{1-4{\beta }_1^2(1+2{\kappa }_0^2)t^2}[\parallel \zeta \parallel +{\parallel U\parallel ]}^2,\forall (\zeta ,U,t)\in \mathbf{B}(K,\delta )}\hfill \end{array}$$

or

$${\displaystyle ∥\left(\begin{array}{c}\widehat{x}(\zeta ,U,t)-\overline{x}\\ \widehat{\pi }(\zeta ,U,t)-\overline{\pi }\\ \widehat{Y}(\zeta ,U,t)-\overline{Y}\end{array}\right)∥\le \frac{2{\beta }_1\sqrt{1+{\kappa }_0^2{t}^2}}{\sqrt{1-4{\beta }_1^2(1+2{\kappa }_0^2)t^2}}[\parallel \zeta \parallel +\parallel U\parallel ],\forall (\zeta ,U,t)\in \mathbf{B}(K,\delta ).}$$

(27)

Define

$$x(\pi ,Y,\sigma )=\widehat{x}(\zeta ,U,t),\tilde{\pi }(\pi ,Y,\sigma )=\widehat{\pi }(\zeta ,U,t),\tilde{Y}(\pi ,U,\sigma )=\widehat{Y}(\zeta ,U,t),\forall (\zeta ,U,t)\in \mathbf{B}(K,\delta ).$$

From the definitions of $D({\sigma }_{*},\delta )$ and K, we have that

$$(\pi ,Y,\sigma )\in D({\sigma }_{*},\delta )⟹(\zeta ,U,t)\in \mathbf{B}(K,\delta )for(\zeta ,U,t)=\left({\displaystyle \frac{\pi -\overline{\pi }}{\sigma },\frac{Y-\overline{Y}}{\sigma },\frac{1}{\sigma }}\right).$$

It follows from (21) that

$$(\overline{x},\overline{\pi },\overline{Y})=(x(\overline{\pi },\overline{Y},\sigma ),\tilde{\pi }(\overline{\pi },\overline{Y},\sigma ),\tilde{Y}(\overline{\pi },\overline{Y},\sigma ))$$

and

$$\begin{array}{c}{\nabla }_x{\mathbb{L}}_{\sigma }(x(\pi ,Y,\sigma ),\pi ,Y)={\nabla }_x\mathbb{L}(x(\pi ,Y,\sigma ),\tilde{\pi }(\pi ,Y,\sigma ),\tilde{Y}(\pi ,Y,\sigma ))=0,\hfill \\ \tilde{Y}(\pi ,Y,\sigma )={\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Y+\sigma g\left(x\right(\pi ,Y,\sigma \left)\right)\right),\tilde{\pi }(\pi ,Y,\sigma )=\pi +\sigma h\left(x(\pi ,Y,\sigma )\right).\hfill \end{array}$$

Noting that $(x(\pi ,Y,\sigma ),\tilde{\pi }(\pi ,Y,\sigma ),\tilde{Y}(\pi ,Y,\sigma ))\in {\mathbf{B}}_{ϵ}(\overline{x},\overline{\pi },\overline{Y})$ and $ϵ<{\delta }_1\le {\delta }_0$ and $\sigma \ge {\sigma }^{*}\ge {\sigma }_0^{*}$ we have from Proposition 1 that

$${\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(x(\pi ,Y,\sigma ),\pi ,Y)\succ 0.$$

Thus, $x(\pi ,U,\sigma )$ is the unique solution of Problem (18) and differentiable on $\mathrm{int}D({\sigma }^{*},\delta )$. Without loss of generality, suppose

$${\sigma }^{*}>\sqrt{{\kappa }_0^2+8{\beta }_1^2(1+2{\kappa }_0^2)}$$

and define $\beta =4{\beta }_1$. The for any $(\pi ,Y,\sigma )\in D({\sigma }^{*},\delta )$, we obtain from (27) that

$${\displaystyle ∥\left(\begin{array}{c}x(\pi ,Y,\sigma )-\overline{x}\\ \widehat{\pi }(\pi ,Y,\sigma )-\overline{\pi }\\ \widehat{Y}(\pi ,Y,\sigma )-\overline{Y}\end{array}\right)∥\le \frac{\beta }{\sigma }\left[\parallel \pi -\overline{\pi }\parallel +\parallel Y-\overline{Y}\parallel \right],}$$

(28)

which implies the estimates (19). □

According to the above theorem, it is easy for us to prove the local convergence properties of the augmented Lagrange method for the nonlinear semidefinite optimization problem.

Proposition 5. 

Let $(\overline{x},\overline{\pi },\overline{\lambda })$ satisfy Jacobian uniqueness conditions. Let $ϵ>0$ and $\delta >0$ be given in Theorem 1. Suppose that ${\sigma }_0>{\sigma }^{*}$, ${\sigma }_k\nearrow {\sigma }_{\infty }$ and $(x^0,{\pi }^0,Y^0)\in {\mathbb{R}}^n\times {\mathbb{R}}^q\times {\mathbb{S}}^p$ satisfy

$${\displaystyle \beta /{\sigma }_0\le 1/4,\parallel x^0-\overline{x}\parallel \le ϵ,\parallel ({\pi }^0,Y^0)-(\overline{\pi },\overline{Y})\parallel \le \delta {\sigma }_0,\delta <\frac{ϵ}{4\beta }.}$$

Then, the sequence $\left\{(x^k,{\pi }^k,Y^k)\right\}$ generated by the ALM is convergent to $(\overline{x},\overline{\pi },\overline{Y})$ with

$${\displaystyle \underset{k\to \infty }{lim\; sup}\frac{\parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel }{\parallel ({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel }\le \frac{\beta }{{\sigma }_{\infty }}}$$

(29)

if ${\sigma }_{\infty }<+\infty$. The sequence $\left\{({\pi }^k,Y^k)\right\}$ converges superlinearly to $(\overline{\pi },\overline{Y})$ when ${\sigma }_{\infty }=+\infty$.

Proof. 

For the sequence $\left\{(x^k,{\pi }^k,Y^k)\right\}$ generated by the ALM, we obtain from Theorem 1 that

$$\begin{array}{c}{\displaystyle \parallel x^1-\overline{x}\parallel \le \frac{\beta }{{\sigma }_0}\parallel ({\pi }^0,Y^0)-(\overline{\pi },\overline{Y})\parallel ,}\hfill \\ {\displaystyle \parallel ({\pi }^1,Y^1)-(\overline{\pi },\overline{Y})\parallel \le \frac{\beta }{{\sigma }_0}\parallel ({\pi }^0,Y^0)-(\overline{\pi },\overline{Y})\parallel ,}\hfill \end{array}$$

which implies

$$\parallel x^1-\overline{x}\parallel \le \beta \delta <ϵ/4$$

and

$${\displaystyle {\sigma }_1^{-1}\parallel ({\pi }^1,Y^1)-(\overline{\pi },\overline{Y})\parallel \le {\sigma }_0^{-1}\parallel ({\pi }^1,Y^1)-(\overline{\pi },\overline{Y})\parallel \le \frac{1}{{\sigma }_0}\frac{\beta }{{\sigma }_0}\left({\sigma }_0\delta \right)\le \frac{1}{{\sigma }_0}\left(\beta \delta \right)\le \frac{\delta }{4}.}$$

Suppose that $(x^k,{\pi }^k,Y^k)$ satisfies ${\sigma }_k^{-1}\parallel ({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel \le \delta$ and $\parallel x^k-\overline{x}\parallel \le ϵ$, then for ${\sigma }_{k+1}\ge {\sigma }_k$, from Theorem 1 we have that

$$\begin{array}{c}{\displaystyle \parallel x^{k+1}-\overline{x}\parallel \le \frac{\beta }{{\sigma }_k}\parallel ({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel ,}\hfill \\ {\displaystyle \parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel \le \frac{\beta }{{\sigma }_k}|({\pi }^k,{\lambda }^k)-(\overline{\pi },\overline{\lambda })\parallel }\hfill \end{array}$$

(30)

which implies

$$\parallel x^{k+1}-\overline{x}\parallel \le \beta \delta <ϵ/4$$

and

$${\displaystyle {\sigma }_{k+1}^{-1}\parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel \le {\sigma }_k^{-1}\parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel \le \frac{1}{{\sigma }_k}\frac{\beta }{{\sigma }_k}\left[\right[|({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel ]\le \frac{\beta }{{\sigma }_k}\delta \le \frac{\delta }{4}.}$$

Therefore, by induction, we obtain that for any $k=1,2,\dots ,$${\sigma }_k^{-1}\parallel ({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel \le \delta$ and $\parallel x^k-\overline{x}\parallel \le ϵ$. Then for $k>1$, we obtain

$$\begin{array}{c}{\displaystyle \parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel \le \frac{\beta }{{\sigma }_k}\times \frac{\beta }{{\sigma }_{k-1}}\times \dots \times \frac{\beta }{{\sigma }_1}\times \frac{\beta }{{\sigma }_0}\parallel ({\pi }^0,Y^0)-(\overline{\pi },\overline{Y})\parallel ,}\hfill \end{array}$$

(31)

which implies

$${\displaystyle \parallel x^{k+1}-\overline{x}\parallel \le \frac{\beta }{{\sigma }_k}\times \frac{\beta }{{\sigma }_{k-1}}\times \dots \times \frac{\beta }{{\sigma }_1}\times \frac{\beta }{{\sigma }_0}\parallel ({\pi }^0,Y^0)-(\overline{\pi },\overline{Y})\parallel .}$$

Noting that $\beta /{\sigma }_0\le 1/4$ and ${\sigma }_k$ is increasing, we obtain from the above inequality that $(x^k,{\pi }^k,Y^k)\to (\overline{x},\overline{\pi },\overline{Y})$. The estimate in (29) comes from (30) and the rate of convergence is superlinear when $c_{\infty }=+\infty$. □

4. Asymptotical Superlinear Convergence of Multipliers

In Theorem 1, the convergence rate of the augmented Lagrange method is characterized by (19), which involves a constant $\beta$. The means by which to give an estimate of $\beta$ are an important topic. In this section, we estimate $\beta$ using the eigenvalues of the second-order derivative of the perturbation function of Problem (1).

Let $(\overline{x},\overline{\pi },\overline{Y})$ be a Kurash–Kuhn–Tucker point of Problem (1), consider the following system of equations in $(x,\pi ,Y,u)$,

$$\begin{array}{cc}\hfill \nabla f\left(x\right)+\mathcal{J}h{\left(x\right)}^T\pi +\mathrm{D}g{\left(x\right)}^{*}Y& \hfill =0,\\ \hfill h\left(x\right)+u_h& \hfill =0,\\ \hfill {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\sigma (g\left(x\right)+u_g)+Y)-Y& \hfill =0,\end{array}$$

(32)

then, $(\overline{x},\overline{\pi },\overline{Y})$ is a solution of (32) for any where $\sigma >0$. According to the implicit function theorem, there exist a constant $\delta >0$ and mappings $(x(·),\pi (·),Y(·))\in {\mathcal{C}}^1\left({\mathbf{B}}_{\delta }\left(0\right)\right)$ such that

$$x\left(0\right)=\overline{x},\pi \left(0\right)=\overline{\pi },Y\left(0\right)=\overline{Y},$$

and for $\parallel u\parallel \le \delta$, where $u=(u_h,u_g)\in {\mathbb{R}}^q\times {\mathbb{S}}^p$,

$$\begin{array}{cc}\hfill \nabla f\left(x\left(u\right)\right)+\mathcal{J}h{\left(x\left(u\right)\right)}^T\pi \left(u\right)+\mathrm{D}g{\left(x\left(u\right)\right)}^{*}Y\left(u\right)& \hfill =0,\\ \hfill h\left(x\left(u\right)\right)+u_h& \hfill =0,\\ \hfill {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\sigma (g\left(x\left(u\right)\right)+u_g)+Y\left(u\right))-Y\left(u\right)& \hfill =0\end{array}$$

(33)

Moreover, there exists $\epsilon >0$ such that

$$x\left(u\right)\in {\mathbf{B}}_{\epsilon }\left(\overline{x}\right),\pi \left(u\right)\in {\mathbf{B}}_{\epsilon }\left(\overline{\pi }\right),Y\left(u\right)\in {\mathbf{B}}_{\epsilon }\left(\overline{Y}\right)$$

for $\parallel u\parallel <\delta$. Define the function $p:{\mathbf{B}}_{\delta }\left(0\right)\to \mathbb{R}$ as follows

$$p\left(u\right)=f\left(x\left(u\right)\right),u\in {\mathbf{B}}_{\delta }\left(0\right).$$

(34)

In view of the Jacobian uniqueness conditions, $\epsilon$ and $\delta$ can be taken small enough so that $x\left(u\right)$ is actually a local minimum point in $\mathrm{int}{\mathbf{B}}_{\epsilon }\left(\overline{x}\right)$ of the following perturbed problem

$$\underset{x\in {\mathbb{R}}^n}{min}\left\{f\left(x\right):h\left(x\right)+u_h=0,g\left(x\right)+u_g⪯0\right\}.$$

(35)

Thus, the function p is actually the following perturbation function:

$$p\left(u\right)=f\left(x\left(u\right)\right)=\underset{x\in {\mathbf{B}}_{\epsilon }\left(\overline{x}\right)}{min}\left\{f\left(x\right):h\left(x\right)+u_h=0,g\left(x\right)+u_g⪯0\right\}.$$

(36)

Lemma 4. 

Suppose that Jacobian uniqueness conditions hold and ε and δ are taken sufficiently small such that $\left(x\right(u),y(u\left)\right)$ is a local minimum point of Problem (35). Then,

$$\mathrm{D}p{\left(u\right)}^{*}=\left(\begin{array}{c}\pi \left(u\right)\\ Y\left(u\right)\end{array}\right),\forall u\in {\mathit{B}}_{\delta }\left(0\right).$$

(37)

Proof. 

We use $\mathcal{L}$ to denote the Lagrange function of Problem (35), namely

$$\mathcal{L}(x,\pi ,Y;u)=f\left(x\right)+{\pi }^T(h\left(x\right)+u_h)+\langle Y,g\left(x\right)+u_g\rangle .$$

Then, $p\left(u\right)$ is expressed as follows

$$p\left(u\right)=f\left(x\left(u\right)\right)+\langle Y\left(u\right),g\left(x\left(u\right)\right)+u_g\rangle +\langle \pi \left(u\right),h\left(x\left(u\right)\right)+u_h\rangle =\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right);u).$$

Noting ${\nabla }_x\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)=0$ and ${\nabla }_{(\pi ,Y)}\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)=0$, from the above expression of $p\left(u\right)$ we obtain

$$\begin{array}{ccc}\hfill \mathrm{D}p{\left(u\right)}^{*}& =& {\mathrm{D}}_{u}x{\left(u\right)}^{*}{\nabla }_x\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)+\mathrm{D}\pi {\left(u\right)}^{*}{\nabla }_{\pi }\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)\hfill \\ & & +\mathrm{D}Y{\left(u\right)}^{*}{\nabla }_Y\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)+{\mathrm{D}}_u\mathcal{L}{(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)}^{*}\hfill \\ & =& {\mathrm{D}}_u\mathcal{L}{(x\left(u\right),\pi \left(u\right),Y\left(u\right);u)}^{*}\hfill \\ & =& \left(\begin{array}{c}\pi \left(u\right)\\ Y\left(u\right)\end{array}\right).\hfill \end{array}$$

The proof is complete. □

Lemma 5. 

Suppose that Jacobian uniqueness conditions hold and ε and δ are taken sufficiently small so that $\left(x\right(u),y(u\left)\right)$ is a local minimum point of the perturbed problem (35). Then,

$$\begin{array}{c}\sigma \left[\begin{array}{cc}{I}_q& 0\\ 0& \mathcal{I}\end{array}\right]+\left[\begin{array}{c}{\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]=\left\{\left[\begin{array}{cc}0& 0\\ 0& {\displaystyle \frac{1}{\sigma }(\mathcal{I}-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right))}\end{array}\right]\right.\hfill \\ {\left.+\left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]{\nabla }_{xx}^2{\mathbb{L}}_{\sigma }{(x\left(u\right),\pi \left(u\right),Y\left(u\right))}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]}^{*}\right\}}^{-1}.\hfill \end{array}$$

(38)

Proof. 

Differentiating (33), we obtain

$${\nabla }_{xx}^2\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right)){\mathrm{D}}_{u}x\left(u\right)+{\mathcal{J}}_{x}h{\left(x\left(u\right)\right)}^T{\mathrm{D}}_u\pi \left(u\right)+{\mathrm{D}}_{x}g{\left(x\left(u\right)\right)}^{*}{\mathrm{D}}_{u}Y\left(u\right)=0$$

(39)

and

$$\begin{array}{c}{\mathcal{J}}_{x}h\left(x\left(u\right)\right){\mathrm{D}}_{u}x\left(u\right)=[-I_{q}0],\hfill \\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\sigma (g\left(x\left(u\right)\right)+u_g)+Y\left(u\right))[\sigma \mathrm{D}g\left(x\left(u\right)\right){\mathrm{D}}_{u}x\left(u\right)+\sigma \left[0\mathcal{I}\right]+{\mathrm{D}}_{u}Y\left(u\right)]-{\mathrm{D}}_{u}Y\left(u\right)=0.\hfill \end{array}$$

(40)

Denote $Z_{\sigma }\left(u\right)=\sigma (g\left(x\left(u\right)\right)+u_g)+Y\left(u\right)$. Then, the Equations (39) and (40) can be written as

$$\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right))\hfill & \mathcal{J}{\left(x\left(u\right)\right)}^T\hfill & \mathrm{D}g{\left(x\left(u\right)\right)}^{*}\hfill \\ \mathcal{J}h\left(x\right(u\left)\right)\hfill & 0\hfill & 0\hfill \\ \sigma \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\hfill & 0\hfill & -\mathcal{I}+\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\hfill \end{array}\right]\left[\begin{array}{c}{\mathrm{D}}_{u}x\left(u\right)\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]=\left[\begin{array}{cc}0& 0\\ -I_q& 0\\ 0& -\sigma \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\end{array}\right]$$

or

$$\left({\overline{\mathbb{K}}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))+\left[\begin{array}{ccc}0& 0& 0\\ 0& {\sigma }^{-1}{I}_q& 0\\ 0& 0& {\sigma }^{-1}\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\end{array}\right]\right)\left[\begin{array}{c}{\mathrm{D}}_{u}x\left(u\right)\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]=\left[\begin{array}{cc}0& 0\\ -I_q& 0\\ 0& -\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\end{array}\right],$$

(41)

where

$${\overline{\mathbb{K}}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))=\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathcal{L}(x\left(u\right),\pi \left(u\right),Y\left(u\right))\hfill & \mathcal{J}h{\left(x\left(u\right)\right)}^T\hfill & \mathrm{D}g{\left(x\left(u\right)\right)}^{*}\hfill \\ \mathcal{J}h\left(x\right(u\left)\right)\hfill & -{\sigma }^{-1}{I}_q\hfill & 0\hfill \\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\hfill & 0\hfill & -{\sigma }^{-1}\mathcal{I}\hfill \end{array}\right].$$

Thus, Equation (41) is equivalent to

$${\overline{\mathbb{K}}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))\left[\begin{array}{c}{\mathrm{D}}_{u}x\left(u\right)\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]=-\left[\begin{array}{ccc}I& 0& 0\\ 0& I_q& 0\\ 0& & \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\end{array}\right]\left\{\left[\begin{array}{cc}0& 0\\ I_q& 0\\ 0& \mathcal{I}\end{array}\right]+{\sigma }^{-1}\left[\begin{array}{c}0\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]\right\}.$$

Therefore, we get that

$$\left[\begin{array}{c}{\mathrm{D}}_{u}x\left(u\right)\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]=-{\sigma }^{-1}{\overline{\mathbb{K}}}_{\sigma }{(x\left(u\right),\pi \left(u\right),Y\left(u\right))}^{-1}\left[\begin{array}{ccc}I& 0& 0\\ 0& I_q& 0\\ 0& 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\end{array}\right]\left\{\sigma \left[\begin{array}{cc}0& 0\\ I_q& 0\\ 0& \mathcal{I}\end{array}\right]+\left[\begin{array}{c}0\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]\right\}$$

which implies

$$\begin{array}{c}\sigma \left[\begin{array}{cc}{I}_q& 0\\ 0& \mathcal{I}\end{array}\right]+\left[\begin{array}{c}{\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]\hfill \\ =-{\sigma }^{-1}\left[\begin{array}{ccc}0& I_q& 0\\ 0& 0& \mathcal{I}\end{array}\right]{\overline{\mathbb{K}}}_{\sigma }{(x\left(u\right),\pi \left(u\right),Y\left(u\right))}^{-1}\left[\begin{array}{ccc}I& 0& 0\\ 0& I_q& 0\\ 0& 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\end{array}\right]\left\{\sigma \left[\begin{array}{cc}0& 0\\ I_q& 0\\ 0& \mathcal{I}\end{array}\right]+\left[\begin{array}{c}0\\ {\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]\right\}\hfill \\ +\sigma \left[\begin{array}{cc}{I}_q& 0\\ 0& \mathcal{I}\end{array}\right].\hfill \end{array}$$

(42)

It follows from Page 20 of [15] that the inverse of ${\overline{\mathbb{K}}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))$ can be expressed as

$$\begin{array}{c}{\overline{\mathbb{K}}}_{\sigma }{(x\left(u\right),\pi \left(u\right),Y\left(u\right))}^{-1}=-\left[\begin{array}{ccc}0& 0& 0\\ 0& \sigma I_q& 0\\ 0& 0& \sigma \mathcal{I}\end{array}\right]\hfill \\ +\left[\begin{array}{c}-I\\ -\sigma \mathcal{J}h\left(x\right(u\left)\right)\\ -\sigma \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]{\left({\overline{\mathbb{K}}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))/\left(\begin{array}{cc}-{\sigma }^{-1}{I}_q& 0\\ 0& -{\sigma }^{-1}\mathcal{I}\end{array}\right)\right)}^{-1}{\left[\begin{array}{c}-I\\ -\sigma \mathcal{J}h\left(x\right(u\left)\right)\\ -\sigma \mathrm{D}g\left(x\right(u\left)\right)\end{array}\right]}^{*}.\hfill \end{array}$$

It is easy to check

$${\overline{\mathbb{K}}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))/\left(\begin{array}{cc}-{\sigma }^{-1}{I}_q& 0\\ 0& -{\sigma }^{-1}\mathcal{I}\end{array}\right)={\nabla }_{xx}^2{\mathbb{L}}_{\sigma }(x\left(u\right),\pi \left(u\right),Y\left(u\right))$$

which implies

$$\begin{array}{c}{\overline{\mathbb{K}}}_{\sigma }{(x\left(u\right),\pi \left(u\right),Y\left(u\right))}^{-1}\hfill \\ =\left[\begin{array}{cc}{\nabla }_{xx}^2{L}_{\sigma }^{-1}(x\left(u\right),\pi \left(u\right),Y\left(u\right))& \sigma {\nabla }_{xx}^2{L}_{\sigma }^{-1}(x\left(u\right),\pi \left(u\right),Y\left(u\right)){\left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}g\left(x\right(u\left)\right)\end{array}\right]}^{*}\\ \sigma \left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]{\nabla }_{xx}^2{L}_{\sigma }^{-1}(x\left(u\right),\pi \left(u\right),Y\left(u\right))& {\Delta }_{\sigma }\left(u\right)\end{array}\right],\hfill \end{array}$$

(43)

where

$${\Delta }_{\sigma }\left(u\right)=-\sigma I+{\sigma }^2\left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]{\nabla }_{xx}^2{L}_{\sigma }^{-1}(x\left(u\right),\pi \left(u\right),Y\left(u\right))\left[\mathcal{J}h{\left(x\left(u\right)\right)}^T\mathrm{D}g{\left(x\left(u\right)\right)}^{*}\right].$$

Therefore, we have from (42) and (43) that

$$\begin{array}{c}\sigma \left[\begin{array}{cc}{I}_q& 0\\ 0& \mathcal{I}\end{array}\right]+\left[\begin{array}{c}{\mathrm{D}}_u\pi \left(u\right)\\ {\mathrm{D}}_{u}Y\left(u\right)\end{array}\right]\hfill \\ ={\left\{\left[\begin{array}{cc}0& 0\\ 0& {\displaystyle \frac{1}{\sigma }(\mathcal{I}-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right))}\end{array}\right]+\left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]{\nabla }_{xx}^2{\mathbb{L}}_{\sigma }{(x\left(u\right),\pi \left(u\right),Y\left(u\right))}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(x\right(u\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(u\right)\right)\mathrm{D}g\left(x\left(u\right)\right)\end{array}\right]}^{*}\right\}}^{-1};\hfill \end{array}$$

namely, the equality (38) holds. □

Corollary 1. 

Let Jacobian uniqueness conditions be satisfied at $(\overline{x},\overline{\pi },\overline{\lambda })$. Then,

$$\begin{array}{c}{\nabla }^{2}p\left(0\right)=-\sigma \left[\begin{array}{cc}{I}_q& 0\\ 0& \mathcal{I}\end{array}\right]\hfill \\ +{\left\{\left[\begin{array}{cc}0& 0\\ 0& {\displaystyle \frac{1}{\sigma }\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{-}^p}\left(Z_{\sigma }^{*}\right)}\end{array}\right]+\left[\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }^{*}\right)\mathrm{D}g\left(\overline{x}\right)\end{array}\right]{\nabla }_{xx}^2{\mathbb{L}}_{\sigma }{(\overline{x},\overline{\pi },\overline{Y})}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }^{*}\right)\mathrm{D}g\left(\overline{x}\right)\end{array}\right]}^{*}\right\}}^{-1},\hfill \end{array}$$

(44)

where $Z_{\sigma }^{*}=Z_{\sigma }\left(0\right)=\overline{Y}+\sigma g\left(\overline{x}\right)$.

Proof. 

The equality (38) is valid for all u with $\parallel u\parallel <\delta$ and all $\sigma$ large enough. For $u=0$, we have

$$\begin{array}{c}\sigma \left[\begin{array}{cc}{I}_q& 0\\ 0& \mathcal{I}\end{array}\right]+\left[\begin{array}{c}{\mathrm{D}}_u\pi \left(0\right)\\ {\mathrm{D}}_{u}Y\left(0\right)\end{array}\right]\hfill \\ ={\left\{\left[\begin{array}{cc}0& 0\\ 0& {\displaystyle \frac{1}{\sigma }(\mathcal{I}-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(0\right)\right))}\end{array}\right]+\left[\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(0\right)\right)\mathrm{D}g\left(\overline{x}\right)\end{array}\right]{\nabla }^2{\mathbb{L}}_{\sigma }{(\overline{x},\overline{\pi },\overline{Y})}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }\left(0\right)\right)\mathrm{D}g\left(\overline{x}\right)\end{array}\right]}^{*}\right\}}^{-1};\hfill \end{array}$$

which implies (44) from (37). □

By using the above properties, we are able to analyze the rate of convergence of multipliers generated by the augmented Lagrange method. For this purpose, we first give an equivalent expression of

$$\left(\begin{array}{c}\tilde{\pi }(\pi ,Y,\sigma )-\overline{\pi }\\ \tilde{Y}(\pi ,Y,\sigma )-\overline{Y}\end{array}\right),$$

which is a key property for analyzing the superlinear rate of the sequence of multipliers.

Theorem 2. 

Let the Jacobian uniqueness conditions be satisfied at $(\overline{x},\overline{\pi },\overline{\lambda })$. Let ${\sigma }^{*}>0$, δ and ε be given by Theorem 1. Then, for all $(\pi ,Y,\sigma )\in D({\sigma }^{*},\delta )$,

$${\displaystyle \left(\begin{array}{c}\tilde{\pi }(\pi ,Y,\sigma )-\overline{\pi }\\ \tilde{Y}(\pi ,Y,\sigma )-\overline{Y}\end{array}\right)={\int }_0^1{{\rm Y}}_{\sigma }(\overline{\pi }+s(\pi -\overline{\pi }),\overline{Y}+s(Y-\overline{Y}))\left(\begin{array}{c}\pi -\overline{\pi }\\ Y-\overline{Y}\end{array}\right)\mathrm{d}s,}$$

(45)

where ${{\rm Y}}_{\sigma }(\pi ,Y)$ is defined by

$$\begin{array}{c}{{\rm Y}}_{\sigma }(\pi ,Y)=\left[\begin{array}{cc}{I}_q& 0\\ 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }(\pi ,Y)\right)\end{array}\right]-\sigma \left[\begin{array}{c}\mathcal{J}h\left(x\right(\pi ,Y,\sigma \left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }(\pi ,Y)\right)\mathrm{D}g\left(x(\pi ,Y,\sigma )\right)\end{array}\right]\times \hfill \\ \times {\nabla }_{xx}^2{L}_{\sigma }{\left(x(\pi ,Y,\sigma ),\tilde{\pi }(\pi ,Y,\sigma ),\tilde{Y}(\pi ,Y,\sigma )\right)}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(x\right(\pi ,Y,\sigma \left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_{\sigma }(\pi ,Y)\right)\mathrm{D}g\left(x(\pi ,Y,\sigma )\right)\end{array}\right]}^{*}\hfill \end{array}$$

(46)

and $Z_{\sigma }(\pi ,Y)=\sigma g\left(x(\pi ,Y,\sigma )\right)+Y$.

Proof. 

Define

$$F_o(x,\tilde{\pi },\tilde{Y};\zeta ,U,t)=\left[\begin{array}{c}\nabla f\left(x\right)+\mathcal{J}h{\left(x\right)}^T\tilde{\pi }+\mathrm{D}g{\left(x\right)}^{*}\tilde{Y}\hfill \\ h\left(x\right)+\zeta +t\overline{\pi }-t\tilde{\pi }\hfill \\ {\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(g\left(x\right)+U+t\overline{Y}\right)-t\tilde{Y}\hfill \end{array}\right].$$

Noting that $F_o(x,\tilde{\pi },\tilde{Y};\zeta ,U,t)=0$ is equivalent to $F(x,\tilde{\pi },\tilde{Y};\zeta ,U,t)$ = 0, we have

$$F_o(\widehat{x}(\zeta ,U,t),\tilde{\pi }(\zeta ,U,t),\tilde{Y}(\zeta ,U,t);\zeta ,U,t)=0.$$

(47)

Differentiating the last three equations in (47) with respect to $(\zeta ,U,t)$, we obtain

$$\begin{array}{c}{\mathrm{D}}_{(x,\tilde{\pi },\tilde{Y})}{F}_o(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t);\zeta ,U,t){\mathrm{D}}_{(\zeta ,U,t)}\left[\begin{array}{c}\widehat{x}(\zeta ,U,t)\\ \widehat{\pi }(\zeta ,U,t)\\ \widehat{Y}(\zeta ,U,t)\end{array}\right]\hfill \\ =\left[\begin{array}{ccc}0& 0& 0\\ -I_q& 0& \widehat{\pi }(\zeta ,U,t)-\overline{\pi }\\ 0& -\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(g\left(\widehat{x}(\zeta ,U,t)\right)+U+t\overline{Y})& \widehat{Y}-\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(g\left(\widehat{x}(\zeta ,U,t)\right)+U+t\overline{Y})\overline{Y}\end{array}\right].\hfill \end{array}$$

(48)

Denoting $A(\zeta ,U,t)={\mathrm{D}}_{(x,\tilde{\pi },\tilde{Y})}{F}_o(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t),t)$ and $Z_o(\zeta ,U,t)=g\left(\widehat{x}(\zeta ,U,t)\right)+U+t\overline{Y}$, we have from (48) that

$$\left[\begin{array}{c}{\mathrm{D}}_{\zeta ,U}\widehat{\pi }(\zeta ,U,t)\\ {\mathrm{D}}_{\zeta ,U}\widehat{Y}(\zeta ,U,t)\end{array}\right]=\left(\begin{array}{ccc}0\hfill & I_q\hfill & 0\hfill \\ 0\hfill & 0\hfill & \mathcal{I}\hfill \end{array}\right)A{(\zeta ,U,t)}^{-1}\left[\begin{array}{cc}0& 0\\ -I_q& 0\\ 0& -\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\end{array}\right].$$

(49)

We can easily obtain the following expression of $A(\zeta ,U,t)$:

$$A(\zeta ,U,t)=\left[\begin{array}{ccc}{\nabla }_{xx}^2\mathcal{L}(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t))\hfill & \mathcal{J}h{\left(\widehat{x}(\zeta ,U,t)\right)}^T\hfill & \mathrm{D}g{\left(\widehat{x}(\zeta ,U,t)\right)}^{*}\hfill \\ \mathcal{J}h\left(\widehat{x}(\zeta ,U,t)\right)\hfill & -t{I}_q\hfill & 0\hfill \\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(\widehat{x}(\zeta ,U,t)\right)\hfill & 0\hfill & -t\mathcal{I}\hfill \end{array}\right].$$

From the equality

$$\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)=\mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(t^{-1}g(x(\zeta ,u,t)+Y)$$

that

$$A{(\zeta ,U,t)}^{-1}=\left[\begin{array}{cc}{\nabla }_{xx}^2{L}_{t^{-1}}{(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t))}^{-1}& {\mathsf{\Theta }}_{12}(\zeta ,U,t)\\ {\mathsf{\Theta }}_{21}(\zeta ,U,t)& {\mathsf{\Theta }}_{22}(\zeta ,U,t)\end{array}\right]$$

(50)

with

$$\begin{array}{c}{\mathsf{\Theta }}_{12}(\zeta ,U,t)=-t^{-1}{\nabla }_{xx}^2{L}_{t^{-1}}{(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t))}^{-1}\left[\mathcal{J}h{\left(\widehat{x}(\zeta ,U,t)\right)}^T\mathrm{D}g\widehat{x}{(\zeta ,U,t)}^{*}\right]\hfill \\ {\mathsf{\Theta }}_{21}(\zeta ,U,t)=-t^{-1}\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(x(\zeta ,U,t)\right)\end{array}\right]{\nabla }_{xx}^2{L}_{t^{-1}}{(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t))}^{-1}\hfill \\ {\mathsf{\Theta }}_{22}(\zeta ,U,t)=-t^{-1}I+t^{-2}\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(x(\zeta ,U,t)\right)\end{array}\right]\times \hfill \\ \times {\nabla }_{xx}^2{L}_{t^{-1}}{(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t))}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}g\left(x\right(\zeta ,U,t\left)\right)\end{array}\right]}^{*}.\hfill \end{array}$$

Thus, we have from (49) that

$$\begin{array}{c}\left[\begin{array}{c}{\mathrm{D}}_{\zeta ,U}\widehat{\pi }(\zeta ,U,t)\\ {\mathrm{D}}_{\zeta ,U}\widehat{Y}(\zeta ,U,t)\end{array}\right]={\mathsf{\Theta }}_{22}(\zeta ,U,t)\left[\begin{array}{cc}{I}_q& 0\\ 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\end{array}\right],\hfill \end{array}$$

(51)

which implies

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\mathrm{D}}_{\zeta ,U}\widehat{\pi }(\zeta ,U,t)\\ {\mathrm{D}}_{\zeta ,U}\widehat{Y}(\zeta ,U,t)\end{array}\right]=t^{-1}\left[\begin{array}{cc}{I}_q& 0\\ 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\end{array}\right]-t^{-2}\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(x(\zeta ,U,t)\right)\end{array}\right]\times \\ \hfill \times {\nabla }^2{L}_{t^{-1}}{\left(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t)\right)}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(x(\zeta ,U,t)\right)\end{array}\right]}^{*}.\end{array}$$

Then, we get

$$\begin{array}{c}\left(\begin{array}{c}\widehat{\pi }(\zeta ,U,t)-\overline{\pi }\\ \widehat{Y}(\zeta ,U,t)-\overline{Y}\end{array}\right)-\left(\begin{array}{c}\widehat{\pi }(\zeta ,U,t)-\pi (0,0,t)\\ \widehat{Y}(\zeta ,U,t)-Y(0,0,t)\end{array}\right)\hfill \\ {\displaystyle =\widehat{\pi }(t,\gamma )-\widehat{\pi }(0,\gamma )={\int }_0^1\left[\begin{array}{c}{\mathrm{D}}_{\zeta ,U}\widehat{\pi }(s\zeta ,sU,t)\\ {\mathrm{D}}_{\zeta ,U}\widehat{Y}(s\zeta ,sU,t)\end{array}\right]\left(\begin{array}{c}\zeta \\ U\end{array}\right)\mathrm{d}s}\hfill \\ {\displaystyle ={\int }_0^1\left\{t^{-1}\left[\begin{array}{cc}{I}_q& 0\\ 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\end{array}\right]-t^{-2}\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(x(\zeta ,U,t)\right)\end{array}\right]\times \right.}\hfill \\ \left.\times {\nabla }^2{L}_{t^{-1}}{\left(\widehat{x}(\zeta ,U,t),\widehat{\pi }(\zeta ,U,t),\widehat{Y}(\zeta ,U,t)\right)}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(x\right(\zeta ,U,t\left)\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Z_o(\zeta ,U,t)\right)\mathrm{D}g\left(x(\zeta ,U,t)\right)\end{array}\right]}^{*}\right\}\left(\begin{array}{c}\zeta \\ U\end{array}\right)\mathrm{d}s.\hfill \end{array}$$

Substituting $t=1/\sigma$, $\zeta =(\pi -\overline{\pi })/\sigma$, $\zeta =(Y-\overline{Y})/\sigma$, $x(\pi ,Y,\sigma )=\widehat{x}(\zeta ,U,t)$, and $\tilde{\pi }(\pi ,Y,\sigma )=\widehat{\pi }(\zeta ,U,t)=\pi -\sigma h\left(x(\zeta ,U,t)\right)$, $\tilde{Y}(\pi ,Y,\sigma )=\widehat{Y}(\zeta ,U,t)={\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(Y+\sigma g\left(\widehat{x}(\zeta ,U,t)\right))$. We obtain the desired result. □

Theorem 3. 

Assume that $(\overline{x},\overline{\pi },\overline{Y})\in {\mathbb{R}}^n\times {\mathbb{R}}^q\times {\mathbb{S}}^p$ satisfies the Jacobian uniqueness y conditions, ${\sigma }^{*}>0$, δ and ε are the constants given by Theorem 1. Suppose that

$${\overline{\sigma }}^{*}>max\left\{{\sigma }^{*},2max\left|\lambda \left({\nabla }^{2}p\left(0\right)\right)\right|\right\}.$$

(52)

Then, there exists a scalar ${\overline{\delta }}_1\in (0,\delta ]$ such that if $({\pi }^0,Y^0)$ and $\left\{{\sigma }_k\right\}$ satisfy

$$∥\left(\begin{array}{c}{\pi }^0-\overline{\pi }\\ Y^0-\overline{Y}\end{array}\right)∥/{\sigma }_0<{\overline{\delta }}_1,{\overline{\sigma }}^{*}\le {\sigma }_k\le {\sigma }_{k+1},\forall k=0,1,2,\dots ,$$

(53)

then the sequence $\left\{({\pi }^k,Y^k)\right\}$ generated by

$$\begin{array}{c}{\pi }^{k+1}={\pi }^k+{\sigma }_{k}h(x({\pi }^k,Y^k,{\sigma }_k),y({\pi }^k,Y^k,{\sigma }_k)),\hfill \\ Y^{k+1}={\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}\left(Y^k+{\sigma }_{k}g(x({\pi }^k,Y^k,{\sigma }_k),y({\pi }^k,Y^k,{\sigma }_k))\right)\hfill \end{array}$$

(54)

is well-defined, and $({\pi }^k,Y^k)\to (\overline{\pi },\overline{Y})$ and $(x({\pi }^k,{\sigma }_k),y({\pi }^k,{\sigma }_k))\to (\overline{x},\overline{Y})$. Furthermore if

$$\underset{k\to \infty }{lim\; sup}{\sigma }_k={\sigma }_{*}<\infty$$

and $({\pi }^k,Y^k)\ne (\overline{\pi },\overline{Y})$ for all k, then

$${\displaystyle \underset{k\to \infty }{lim\; sup}\frac{\parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel }{\parallel ({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel }\le max\left|{\displaystyle \frac{\lambda \left({\nabla }^{2}p\left(0\right)\right)}{{\sigma }_{*}+\lambda \left({\nabla }^{2}p\left(0\right)\right)}}\right|,}$$

(55)

while if ${\sigma }_k\to \infty$ and $({\pi }^k,Y^k)\ne (\overline{\pi },\overline{Y})$ for all k, then

$${\displaystyle \underset{k\to \infty }{lim}\frac{\parallel ({\pi }^{k+1},Y^{k+1})-(\overline{\pi },\overline{Y})\parallel }{\parallel ({\pi }^k,Y^k)-(\overline{\pi },\overline{Y})\parallel }=0.}$$

(56)

Proof. 

In view of ${\mathsf{{\rm Y}}}_{\sigma }$ of (46), we have that

$$\begin{array}{c}{\mathsf{{\rm Y}}}_{\sigma }(\overline{\pi },\overline{Y})=\left[\begin{array}{cc}{I}_q& 0\\ 0& \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\sigma g\left(\overline{x}\right)+\overline{Y})\end{array}\right]\hfill \\ -\sigma \left[\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\sigma g\left(\overline{x}\right)+\overline{Y})\mathrm{D}g\left(\overline{x}\right)\end{array}\right]{\nabla }^2{L}_{\sigma }{\left(\overline{x},\overline{\pi },\overline{Y}\right)}^{-1}{\left[\begin{array}{c}\mathcal{J}h\left(\overline{x}\right)\\ \mathrm{D}{\mathsf{\Pi }}_{{\mathbb{S}}_{+}^p}(\sigma g\left(\overline{x}\right)+\overline{Y})\mathrm{D}g\left(\overline{x}\right)\end{array}\right]}^{*}.\hfill \end{array}$$

Using (44), we obtain

$${\mathsf{{\rm Y}}}_{\sigma }(\overline{\pi },\overline{Y})=I-\sigma {({\nabla }^{2}p\left(0\right)+\sigma I)}^{-1}$$

and thus for an eigenvalue $\lambda \left({\mathsf{{\rm Y}}}_{\sigma }\right))$, one has

$${\displaystyle \lambda \left({\mathsf{{\rm Y}}}_{\sigma }(\overline{\pi },\overline{Y})\right)=\frac{\lambda \left({\nabla }^{2}p\left(0\right)\right)}{\sigma +\lambda \left({\nabla }^{2}p\left(0\right)\right)},}$$

where $\lambda \left({\nabla }^{2}p\left(0\right)\right)$ denotes the corresponding eigenvalue of ${\nabla }^{2}p\left(0\right)$. It is obvious that for any ${\epsilon }_1>0$, there exists ${\overline{\delta }}_1\in (0,\delta ]$ such that for all $(\pi ,Y,\sigma )$ with $\sigma \ge {\sigma }^{*}$ and $\parallel (\pi -\overline{\pi },Y-\overline{Y})\parallel /\sigma <{\overline{\delta }}_1$ we have

$${\displaystyle \parallel {\mathsf{{\rm Y}}}_{\sigma }(\pi ,Y)\parallel \le \parallel {\mathsf{{\rm Y}}}_{\sigma }(\overline{\pi },\overline{Y})\parallel +{\epsilon }_1=max\left|\lambda \left({\mathsf{{\rm Y}}}_{\sigma }(\overline{\pi },\overline{Y})\right)\right|+{\epsilon }_1=max\left|\frac{\lambda \left({\nabla }^{2}p\left(0\right)\right)}{\sigma +\lambda \left({\nabla }^{2}p\left(0\right)\right)}\right|+{\epsilon }_1,}$$

where $\parallel ·\parallel$ denotes the spectral norm of the operator. Using (45) for all $(\pi ,Y,\sigma )$ chosen as in the above, we have

$$∥\left(\begin{array}{c}\tilde{\pi }(\pi ,Y,\sigma )-\overline{\pi }\\ \tilde{Y}(\pi ,Y,\sigma )-\overline{Y}\end{array}\right)∥\le \left({\displaystyle max\left|\frac{\lambda \left({\nabla }^{2}p\left(0\right)\right)}{\sigma +\lambda \left({\nabla }^{2}p\left(0\right)\right)}\right|+{\epsilon }_1}\right)∥\left(\begin{array}{c}\pi -\overline{\pi }\\ Y-\overline{Y}\end{array}\right)∥.$$

(57)

From (52) and (53), we have

$${\displaystyle max\left|{\displaystyle \frac{\lambda \left({\nabla }^{2}p\left(0\right)\right)}{\sigma +\lambda \left({\nabla }^{2}p\left(0\right)\right)}}\right|<1.}$$

Thus, by choosing a sufficiently small ${\epsilon }_1$, we can determine that there exists ${\varrho }_1\in (0,1)$ such that

$$∥\left(\begin{array}{c}\tilde{\pi }(\pi ,Y,\sigma )-\overline{\pi }\\ \tilde{Y}(\pi ,Y,\sigma )-\overline{Y}\end{array}\right)∥\le {\varrho }_1∥\left(\begin{array}{c}\pi -\overline{\pi }\\ Y-\overline{Y}\end{array}\right)∥$$

for $(\pi ,Y,\sigma )$ with $\sigma \ge {\overline{\sigma }}^{*}$ and $\parallel (\pi ,Y)-(\overline{\pi },\overline{Y})\parallel /\sigma <{\delta }_1$. Combining this with (19) and (53), we obtain that ${\pi }^k\to \overline{\pi }$ and $(x({\pi }^k,{\sigma }_k),y({\pi }^k,{\sigma }_k))\to (\overline{x},\overline{Y})$. The estimates (55) and (56) for the convergence rate can be easily obtained directly from (57). □

5. Conclusions

In this paper, we have studied the convergence rate of the augmented Lagrangian method for the nonlinear semidefinite optimization problem. We have proven the local linear rate of convergence of the sequence of multipliers and that the ratio constant is proportional to $1/\sigma$ when $\sigma$ exceeds a threshold ${\sigma }^{*}>0$ and the ratio $\parallel (\pi ,Y)-({\pi }^{*},Y^{*})\parallel /\sigma$ is sufficiently small. Importantly, based on the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we have obtained an accurate estimation for the rate constant of the linear convergence of multiplier vectors generated by the augmented Lagrange method, which shows that the sequence of multipliers is superlinear convergent if $\left\{{\sigma }_k\right\}$ is increasing to ∞.

There are many unsolved problems left in the augmented Lagrange method for nonlinear semidefinite optimization problems. First, in Theorem 1, the result on the convergence rate of the augmented Lagrange method is obtained when the subproblems are exactly solved. A natural problem is how to analyze the convergence rate of the ALM when the subproblems are solved inexactly. Second, all results in this paper are about local convergence of the augmented Lagrange method, global convergent augmented Lagrangian methods are worth studying. Third, for estimating the rate constant of linear convergence, we need the strict complementarity condition; this is a critical condition. What about the convergence properties of the augmented Lagrange method when this condition does not hold?