# Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations
## 1 Introduction  
Let \(F:R^{n}\to R^{n}\) be locally Lipschitz continuous. Consider the nonlinear equation in the following simplest form:  
\[F(x)=0, \tag{1}\]  
where \(x=(x_{1},\ldots,x_{n})^{T}\), \(F(x)=(f_{1}(x),\ldots,f_{n}(x))^{T}\) and \(f_{i}:R^{n}\to R\), \(i=1,\ldots,n\). Furthermore, we assume that there exists \(x^{*}\in R^{n}\), which is a solution to Equation (1) (i.e., \(F(x^{*})=0\)). It is worth recalling that [PERSON] and [PERSON] [1] presented significant motivations and a broad scope of applications for nonsmooth equations. Some class of iterative exponential methods for solving one-dimensional nonlinear equations was proposed by [PERSON] and [PERSON] [2]. One of the most interesting formulas, and simultaneously the simplest one, is the method which has the following form:  
\[x_{k+1}=x_{k}\exp\biggl{\{}-\frac{f(x_{k})}{x_{k}f^{\prime}(x_{k})}\biggr{\}},\;k=0,1,2,\ldots \tag{2}\]  
This method turns to the classic [PERSON] method if we use the first-order Taylor series expansion of the expression \(\exp\Bigl{\{}-\frac{f(x_{k})}{x_{k}f(x_{k})}\Bigr{\}}\). The above method in Equation (2) has (at least) quadratic convergence if the function \(f:R\to R\) is twice differentiable and \(f^{\prime}(x)\
eq 0\) in a neighborhood of the solution. A substantial extension of the exponential method in Equation (2) to the nonsmooth case in \(R^{n}\) was presented in [3]. This locally and superlinearly convergent method can be written in the following two-stage form:  
\[\left\{ \begin{array}{l}V_{k}h^{(k)}=-F(x^{(k)})\\ x^{(k+1)}=\mbox{diag}\biggl{(}\exp\left\{\frac{h^{(k)}_{i}}{x^{(k)}_{i}} \right\}\biggr{)}\biggr{)}x^{(k)}\end{array} \right. \tag{3}\]  
where the matrix \(V_{k}\) is arbitrarily taken from the B-differential of \(F\) at \(x^{(k)}\). On the other hand, one traditional approach for solving nonsmooth equations is the inexact generalized [PERSON] method:\[\left\|F(x^{(k)})+V_{k}h^{(k)}\right\|\leq\eta_{k}\left\|F(x^{(k)})\right\|, \tag{4}\]  
where the matrix \(V_{k}\) could be arbitrarily taken from some subdifferential of \(F\) at \(x^{(k)}\) (not only from the generalized Jacobian or B-differential but also from, for example, the *-differential) and \(0\leq\eta_{k}<1\) for all \(k=0,1,2,\ldots\) [PERSON] and [PERSON] in [4] stated the local and superlinear convergence of the generalized Jacobian-based method with Equation (4) for solving equations with semismooth function \(F\). Other versions of such Newton-type methods for solving nonsmooth equations were also considered among the others in [5; 6; 7]. The main concept of the inexact Newton-type method is to substitute the standard Newton equation \(V_{k}h^{(k)}=-F(x^{(k)})\) with an inequality as in Equation (4), where the parameter \(\eta_{k}\in[0,1)\) is the forcing term. Then, the Newton step \(h_{k}\) may be determined inexactly, such as by the use of any iterative method with feasible inaccuracy, especially when the current approximation is far from the solution to Equation (1). The main objective of this paper is to introduce an inexact version of the exponential method with a subdifferential for multidimensional nonsmooth cases. Therefore, we present some new iterative methods for solving nondifferentiable problems, which can be sufficiently effective in solving not only nonsmooth equations but also some important problems in nonsmooth optimization. The exponential method was introduced in [2] for finding single roots of univariate nonlinear equations and considered in [8] for solving unconstrained optimization problems. In both cases, the proposed formulas allow solving only smooth and one-dimensional problems. We discuss an algorithm which is intended to solve both nonsmooth and multi-dimensional problems. Our method allows solving Equation (1), in which the function is only Lipschitz continuous and need not be differentiable. A primary extension, shown in Equation (3), was introduced in [3], but this study focuses on the inexact approach.