# Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations
## 3 The Algorithm and Its Properties  
In this section, we introduce a new algorithm, and we discuss its convergence. For solving Equation (1), we can consider a method in the descriptive two-stage form  
\[\left\{ \begin{array}{l}\text{determine }h^{(k)}\text{ using }V_{k}\text{ and }F(x^{(k)})\\ \text{compute }x^{(k+1)}\text{ using }x^{(k)}\text{ and }h^{(k)}\end{array} \right.\]  
where matrix \(V_{k}\) is taken from some subdifferential of \(F\) at \(x^{(k)}\). We suggest an approach which is a combination of two methods: the iterative exponential one and the inexact generalized Newton one with a decreasing function norm and backtracking. The proposed inexact approach is similar to the algorithm presented by [PERSON] et al. [17] for smooth equations, whereas the exponential formula was introduced in [3]. Because of the nondifferentiability of \(F\), as iteration matrices, we use matrices from the B-differential instead of the Jacobians. Remark 2:
1. _The matrix_ \(V_{k}\) _may be chosen absolutely arbitrarily from known elements of the Bouligand subdifferential (in Step_ 1 _of Algorithm_ 1_). Appropriate matrices are usually suitable Jacobians of_ \(F\)_._ 2. _The backtracking approach used in Step_ 2 _is a particular case of the minimum reduction algorithm._ 3. _The formula in Equation (_10_) for obtaining_ \(x^{(k+1)}\) _could be written in a substitute form, which explicitly shows how to generate a new approximation_ \(x^{(k+1)}\) _using components of the previous approximation_ \(x^{(k)}\)_:_ \[x^{(k+1)}_{i}=x^{(k)}_{i}\exp\left\{\frac{\alpha_{k}h^{(k)}_{i}}{x^{(k)}_{i}} \right\},\ i=1,\ldots,n.\] (7)
2. _Various strategies for the choice of the forcing sequence can be found in_ _[_18_]_ _but for the inexact methods for solving smooth equations. Therefore, most of these strategies would need to be modified with respect to the nondifferentiability of_ \(F\)_. For example, we could set_ \(\eta_{k}\in[0,1)\) _as a small constant or use the following rule:_\[\eta_{k}=\left\{ \begin{array}{ll}\frac{\|V_{k}h^{(k)}+F(x^{(k)})\|}{\|F(x^{(k)}) \|}&\text{ if }F(x^{(k)})\
eq 0\\ 0&\text{ otherwise,}\end{array} \right.\]  
_where \(V_{k}\in\partial_{B}F(x^{(k)})\). Then, Equation (8) holds with equality._  
Before the convergence of Algorithm 1 is proven, we present a necessary important assumption:  
```
Assume that \(\theta\in(0,1)\), \(\tau_{1},\tau_{2}\in(0,1)\) and \(\tau_{1}<\tau_{2}\). Let \(x^{(0)}\in R^{n}\) be an arbitrary starting approximation. The steps for obtaining \(x^{(k+1)}\) from \(x^{(k)}\) are as follows: Step 1: Determine \(\eta_{k}\in[0,1)\) and find an \(h^{(k)}\) value that satisfies \[\left\|V_{k}h^{(k)}+F(x^{(k)})\right\|\leq\eta_{k}\Big{\|}F(x^{(k)})\Big{\|},\] (8) where \(V_{k}\in\partial_{B}F(x^{(k)})\). Set \(\alpha=1\). Step 2: If \[\left\|F\bigg{(}\text{diag}\bigg{(}\exp\left\{\frac{\alpha h_{i}^{(k)}}{x_{i} ^{(k)}}\right\}\bigg{)}x^{(k)}\bigg{)}\right\|\leq[1-\alpha\theta(1-\eta_{k}) ]\Big{\|}F(x^{(k)})\Big{\|}\] (9) then set \(\alpha_{k}=\alpha\) and \[x^{(k+1)}=\text{diag}\bigg{(}\exp\left\{\frac{\alpha_{k}h_{i}^{(k)}}{x_{i}^{ (k)}}\right\}\bigg{)}x^{(k)}.\] (10) Otherwise, choose \(\alpha^{\prime}\in[\tau_{1}\alpha,\tau_{2}\alpha]\), set \(\alpha=\alpha^{\prime}\), and repeat Step 2. ```  
**Algorithm 1**Inexact exponential method (IEM). **Assumption 1**. : _Function \(F\) satisfies Assumption 1 at \(x\) if for any \(y\) from a neighborhood \(N\) of \(x\) and any matrix \(V_{y}\in\partial_{B}F(y)\), it holds that_  
\[F(y)-F(x)=V_{y}(y-x)+o(\|y-x\|).\]  
_Furthermore, function \(F\) satisfies Assumption 1 at \(x\) with a degree \(\rho\) if the following holds:_  
\[F(y)-F(x)=V_{y}(y-x)+O(\|y-x\|^{\rho}).\]  
**Remark 3**. :
1. _It is not difficult to indicate functions that satisfy Assumption_ 1_. We can indicate three classes of such functions. Assumption_ 1 _is implied not only by semismoothness but also by_ \(H\)_-differentiability (introduced by [PERSON] et al._ _[_19_]__) and second-order_ \(C\)_-differentiability (introduced by [PERSON]_ _[_10_]__)._ 2.