# Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations
## 2 Preliminaries  
Throughout the paper, \(\|\cdot\|\) denotes the Euclidean norm on \(R^{n}\), where we regard the vector from \(R^{n}\) as a column vector. However, all theoretical results do not depend on the choice of the norm. We assume that the function \(F\) is locally Lipschitz continuous in the traditional sense. In other words, for every \(U\subset R^{n}\), there exists a constant \(L>0\) such that  
\[\|F(x)-F(y)\|\leq L\|x-y\|\text{ for all }x,y\in U.\]  
If \(F:R^{n}\to R^{n}\) is a differentiable function, we denote the Jacobian matrix of \(F\) at \(x\) as \(JF(x)\) and the set where \(F\) is differentiable as \(D_{F}\). The local Lipschitz continuity of the function \(F\) implies that it is differentiable almost everywhere, according to [PERSON]'s theorem [9]. Then, we have  
\[\partial_{B}F(x)=\left\{\lim_{x_{i}\to x}JF(x_{i}),x_{i}\in D_{F}\right\}\]  
which is called the Bouligand subdifferential (B-differential for short) of \(F\) at \(x\)[10]. Furthermore, the generalized Jacobian of function \(F\) at \(x\) (in the sense of [PERSON] [9]) is the convex hull of the Bouligand subdifferential such that  
\[\partial F(x)=\text{conv}\partial_{B}F(x).\]We say that the function \(F\) is BD-regular at \(x\) if \(F\) is locally Lipschitz continuous at \(x\) and if all matrices \(V\) from \(\partial_{B}F(x)\) are nonsingular. For BD-regular functions, Qi proved some important properties (see Lemma 2). In turn, the set  
\[\partial_{b}F(x)=\partial_{B}f_{1}(x)\times\ldots\times\partial_{B}f_{n}(x),\]  
is called the b-differential of function \(F\) at \(x\) (defined by [PERSON] and [PERSON] in [11]). It is easy to verify that in a one-dimensional case (i.e., when \(n=1\)), the generalized Jacobian reduces to the Clarke generalized gradient and \(\partial_{B}F(x)=\partial_{b}F(x)\) i.e., the B-differential and b-differential coincide). Furthermore, a \(*\)-differential \(\partial_{*}F(x)\) is a non-empty bounded set for any \(x\) such that  
\[\partial_{*}F(x)\subset\partial f_{1}(x)\times\ldots\times\partial f_{n}(x),\]  
where \(\partial f_{i}(x)\) denotes the generalized gradient of the component function \(f_{i}\) at \(x\). Obviously, for a locally Lipschitz function, all previously mentioned differentials, namely \(\partial F(x)\), \(\partial_{B}F(x)\) and \(\partial_{b}F(x)\), are \(*\)-differentials (see Gao [12]). Obviously, if all components \(f_{i}\) of function \(F\) are \(C^{1}\) at \(x\), then \(\partial F(x)=\partial_{B}F(x)=\{JF(x)\}\). The notion of semismoothness was primarily introduced by [PERSON] [13] for functionals. Later, [PERSON] and [PERSON] [14] extended the original definition to nonlinear operators. A function \(F\) is semismooth at \(x\) if \(F\) is locally Lipschitz continuous at \(x\) and for any \(h\in R^{n}\), the limit  
\[\lim_{V\in\partial F(x+h^{\prime}),h^{\prime}\to h,t\downarrow 0}Vh^{\prime}.\]  
exists. If function \(F\) is semismooth at \(x\), then \(F\) is directionally differentiable at \(x\), and \(F^{\prime}(x;h)\) is equal to the above limit:  
Lemma 1 (Lemma 2.2 by [PERSON] [15]): _Suppose that the function \(F:R^{n}\to R^{n}\) is directionally differentiable in a neighborhood of \(x\). The following statements are equivalent:_  
1. \(F\) _is semismooth at_ \(x\)_;_
2. \(F^{\prime}(\cdot,\cdot)\) _is semicontinuous at_ \(x\)_;_
3. _For any_ \(V\in\partial F(x+h)\)_, the following is true:_ \[Vh-F^{\prime}(x;h)=o(\|h\|)\text{ as }h\to 0.\]  
Moreover, we say that function \(F\) is \(p\)-order semismooth at \(x\) if for any matrix \(V\) from the generalized Jacobian \(\partial F(x+h)\), it holds that  
\[Vh-F^{\prime}(x,h)=O(\|h\|^{1+p})\text{ as }h\to 0,\]  
where \(0<p\leq 1\). If \(p=1\), then \(F\) is just called strongly semismooth [16]. Remark 1:
1. Clearly, semismoothness of any order implies semismoothness. 2. The piecewise \(C^{2}\) function is strongly semismooth.