## 3 The Feasible Nonsmooth Levenberg-Marquardt-Type Method and Its Convergence Analysis
Compute \(t_{k}=\max\{\beta^{\prime}|l=0,1,2\cdots\}\), such that:_  
\[\theta(z_{k}(t_{k}))\leq\theta(z_{k})+\sigma\
abla\theta(z_{k})^{T}(z_{k}(t_{k })-z_{k}),\]  
_where \(z_{k}(t_{k})=P_{Z}[z_{k}-t_{k}\
abla\theta(z_{k})]\). Set \(z_{k+1}=z_{k}(t_{k})\), \(k=k+1\), and go to Step 1._  
We now investigate the convergence properties of Method 1. In the following sections, we assume that Method 1 generates an infinite sequence. **Theorem 1**. : _Method 1 is well defined for a monotone SLCP (3). If Method 1 does not stop at a stationary point in finite steps, an infinite sequence \(\{z_{k}\}\) is generated with \(\{z_{k}\}\subset Z\), and any accumulation point of the sequence \(\{z_{k}\}\) is a stationary point of \(\theta\)._  
**Proof of Theorem 1**. : Method 1 is well defined for the reason of \(\
u_{k}>0\), and \(d_{k}\) is always a descent direction for \(\theta\). Now, we consider the following two situations respectively. * If the direction \(d_{k}\) is accepted by an infinite number of times in Step 3 of Method 1, we get: \[z_{k+1}=P_{Z}[z_{k}+d_{k}]\in Z.\] Since \(\
abla\theta(z_{k})\
eq 0\) implies \(d_{k}\
eq 0\), we have: \[\
abla\theta(z_{k})^{T}d_{k}=-((H_{k}^{T}H_{k}+\
u_{k}I)d_{k})^{T}d_{k}<0.\] From [17], we know that \(\{\theta(z_{k})\}\) is monotonically decreasing. Obviously, this implies that the sequence \(\{\|F(z_{k})\|\}\) is also monotonically decreasing. Since \(\|F(P_{Z}(z_{k}+d_{k}))\|\leq\gamma\|F(z_{k})\|\) is accepted by an infinite number of times in view of our assumptions, therefore we get \(\|F(z_{k})\|\to 0\) for \(k\to\infty\) by \(\gamma\in(0,1)\). This means that any accumulation point of \(\{z_{k}\}\) is the solution of (10); therefore, it is also a stationary point of \(\theta\). * This case is the negation of Case (I); without loss of generality, we assume that the Levenberg-Marquardt direction is never accepted. If the direction \(P_{Z}[z_{k}-t_{k}\
abla\theta(z_{k})]-z_{k}\) is accepted by an infinite number of times in Step 4 of Method 1, we have: \[z_{k+1}=P_{Z}[z_{k}-t_{k}\
abla\theta(z_{k})]\in Z.\] By (b) in Lemma 1, taking \(x:=z_{k}-t_{k}\
abla\theta(z_{k})\), \(y:=z_{k}\), we get: \[0 \geq [P_{Z}(z_{k}-t_{k}\
abla\theta(z_{k}))-(z_{k}-t_{k}\
abla\theta(z _{k}))]^{T}[P_{Z}(z_{k}-t_{k}\
abla\theta(z_{k}))-z_{k}]\] \[s.t. x\geq 0,y_{1}\geq 0,y_{2}\geq 0.\]0 \[s.t. x\geq 0,y_{1}\geq 0,y_{2}\geq 0.\]1 that is, \[s.t. x\geq 0,y_{1}\geq 0,y_{2}\geq 0.\]2 where \(t_{k}=\max\{\beta^{l}|l=0,1,2\cdots\}\) with \(\beta\in(0,1)\). By the _Armijo_ line search properties, we know that any accumulation point of \(\{z_{k}\}\) is a stationary point of \(\theta\), and this completes the proof. **Theorem 2**. : _Let \(x^{*}\in\Re^{n}\) be a \(R-\)regular solution; then the whole sequence generated by Method 1 converges to \(z^{*}\)\(Q\)-quadratically._  
Proof of Theorem 2.: By Proposition 6, there is a constant \(c_{1}>0\), such that, for all \(z_{k}\in\bigcup(z^{*},\delta_{1})\), where \(\delta_{1}\) is a sufficiently small positive constant, the matrices \(H_{k}^{T}H_{k}+v_{k}I\) are nonsingular, and \(\|(H_{k}^{T}H_{k}+v_{k}I)^{-1}\|\leq c_{1}\) hold. Furthermore, by Proposition 2, there exists a constant \(c_{2}>0\), such that:  
\[s.t. x\geq 0,y_{1}\geq 0,y_{2}\geq 0.\]3  
for all \(z_{k}\in\bigcup(z^{*},\delta_{2})\), where \(\delta_{2}\) is a sufficiently small positive constant. Moreover, in view of the upper semicontinuity of the C-subdifferential, we have:  
\[s.t. x\geq 0,y_{1}\geq 0,y_{2}\geq 0.\]4  
where \(H_{k}\in\partial_{C}F(z_{k})\), \(\zeta>0\), \(z_{k}\in\bigcup(z^{*},\delta_{3})\), and \(\delta_{3}\) is a sufficiently small positive constant. Denote \(\delta=min(\delta_{1},\delta_{2},\delta_{3})\), for \(z_{k}\in\bigcup(z^{*},\delta)\).