# Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations
## 3 The Algorithm and Its Properties
\]In addition, since  
\[\|h\|=\frac{1-\eta}{1-\eta}\|\bar{h}\|\leq\frac{1-\eta_{\min}}{1-\bar{\eta}}\| \bar{h}\|\leq\delta,\]  
then  
\[ \begin{array}{ll}&\left\|F\Big{(}\text{diag}\Big{(}\exp\Big{\{}\frac{ah_{1}}{x_ {i}}\Big{\}}\Big{)}x\Big{)}\right\|\\ &\leq\ \left\|F\Big{(}\text{diag}\Big{(}\exp\Big{\{}\frac{ah_{1}}{x_{i}}\Big{\}} \Big{)}x\Big{)}-F(x)-Vh\right\|+\|Vh+F(x)\|\\ &\leq\ \epsilon\frac{1-\eta}{1-\eta}\|\bar{h}\|+\eta\|F(x)\|\\ &=\ (1-\alpha\theta)(1-\eta)\|F(x)\|+\eta\|F(x)\|\\ &=\ [1-\alpha\theta(1-\eta)]\|F(x)\|.\end{array} \]  
\(\square\)  
Lemma 4 above implies that the inexact exponential step is well-defined for the current approximation \(x^{(k)}\), and suitable \(\eta_{k}\) and \(h^{(k)}\) values exist for any \(\theta\in(0,1)\). Now, we will prove that the method described by Algorithm 1 is convergent. The first theorem regarding convergence is analogous to the one presented in [20] for the inexact generalized Newton method for solving nonsmooth equations, albeit with some significant changes in the proof:  
Theorem 4.1: _Let \(\{x^{(k)}\}\) be a sequence generated by Algorithm 1 with \(\eta_{k}\to 0\), and for each \(k\), there is a constant \(M>0\) such that the following holds:_  
\[\left\|V_{k}h^{(k)}+F(x^{(k)})\right\|\leq\eta_{k}\Big{\|}F(x^{(k)})\Big{\|}\]  
_and_  
\[\left\|h^{(k)}\right\|\leq M\]  
_for any matrix \(V_{k}\in\partial_{B}F(x^{(k)})\)._  
_If function \(F\) satisfies Assumption 1 at \(x^{*}\), then every limiting point of the sequence \(\{x^{(k)}\}\) is a solution to Equation (1), and_  
\[\lim_{k\to\infty}F(x^{(k)})=0.\]  
Proof of Theorem 4.1.: Suppose that \(K\) is a sequence of indices such that \(\lim_{k\in K}x^{(k)}=x^{*}\). We have two cases due to the behavior of the sequence \(\{\alpha_{k}\}\):  
(i) \(\{\alpha_{k}\}\) is not convergent to \(0\). Then, there is a sequence of indices \(K_{0}\subset K\) and \(\bar{\alpha}>0\) such that \(\alpha_{k}\geq\bar{\alpha}>0\) for all \(k\in K_{0}\). Clearly, for all \(k\
otin K_{0}\), the following is true:  
\[\left\|F(x^{(k+1)})\right\|\leq\left\|F(x^{(k)})\right\|.\]  
In turn, according to Equation (9), we have  
\[\left\|F(x^{(k+1)})\right\|\leq\left\|F(x^{(k)})\right\|-\bar{\alpha}\theta(1 -\eta_{k})\left\|F(x^{(k)})\right\|\text{ for all }k\in K_{0}.\]  
By adding all these inequalities side by side, we obtain the following inequality:  
\[\bar{\alpha}\theta\sum_{k\in K_{0}}(1-\eta_{k})\left\|F\Big{(}x^{(k)}\Big{)} \right\|\leq\left\|F\Big{(}x^{(0)}\Big{)}\right\|.\]  
Therefore, \(\lim_{k\in K_{0}}F(x^{(k)})=0\). Hence, \(F(x^{*})=0\). (ii) \(\{\alpha_{k}\}\) is convergent to \(0\). Since both \(\eta_{k}\to 0\) and \(\left\|F(x^{(k)})\right\|\to 0\), we obtain  
\[\lim_{k\to\infty}\frac{\left\|F\bigg{(}\text{diag}\bigg{(}\exp\bigg{\{}\frac{ \alpha_{k}h_{i}^{(k)}}{x_{i}^{(k)}}\bigg{\}}\bigg{)}x^{(k)}\bigg{)}\right\|}{ \left\|F\big{(}x^{(k)}\big{)}\right\|}=0.\]  
Thus, for \(k\) values that are large enough, we have  
\[\left\|F\bigg{(}\text{diag}\bigg{(}\exp\bigg{\{}\frac{\alpha_{k}h_{i}^{(k)}}{x_ {i}^{(k)}}\bigg{\}}\bigg{)}x^{(k)}\bigg{)}\right\|\leq(1-\theta)\left\|F(x^{(k) })\right\|.\]  
Therefore, the inequality in Equation (9) holds with \(\alpha=1\), and for \(k\) values that are large enough, we obtain  
\[x^{(k+1)}=\text{diag}\bigg{(}\exp\bigg{\{}\frac{\alpha_{k}h_{i}^{(k)}}{x_{i}^{ (k)}}\bigg{\}}\bigg{)}x^{(k)}.\]  
Hence, we have  
\[\lim_{k\to\infty}\frac{\left\|F(x^{(k+1)})\right\|}{\left\|F(x^{(k)})\right\| }=0. \tag{14}\]  
Now, under the local Lipschitz continuity and BD-regularity of function \(F\) at \(x^{*}\) and Assumption 1, there exists \(l>0\) such that  
\[\left\|h^{(k)}\right\|\leq M\]0  
for all \(x\) in a neighborhood of \(x^{*}\) (see Remark 3 (iii)). Then, according to Equation (14), we have  
\[\left\|h^{(k)}\right\|\leq M\]1  
**Remark 4**. :
1. _[label=()]_
2. _It is not hard to check that Theorem_ 2 _is still true if our new Algorithm_ 1 _is used with one of other subdifferentials mentioned in Section_ 1_._
3.