# On a Nonsmooth Gauss-Newton Algorithms for Solving Nonlinear Complementarity Problems
## 2 Materials and Methods  
### Preliminaries  
If \(F\) is Lipschitz continuous, the [PERSON]'s theorem [14] implies that \(F\) is almost everywhere differentiable. Let the set of points, where \(F\) is differentiable, be denoted by \(D_{F}\). Then, the B-differential (the Bouligand differential) of \(F\) at \(\mathbf{x}\) (introduced in [15]) is  
\[\partial_{B}F(\mathbf{x})=\left\{\lim_{\mathbf{x}^{(n)}\to\mathbf{x}}F^{\prime}\left(\mathbf{ x}^{(n)}\right),\mathbf{x}^{(n)}\in D_{F}\right\},\]where \(F^{\prime}(\mathbf{x})\) denotes the usual Jacobian of \(F\) at \(\mathbf{x}\). The generalized Jacobian of \(F\) at \(\mathbf{x}\) in the sense of [PERSON] [14] is  
\[\partial F(\mathbf{x})=conv\partial_{B}F(\mathbf{x})\]  
We say that \(F\) is BD-regular at \(\mathbf{x}\), if \(F\) is locally Lipschitz at \(\mathbf{x}\) and if all \(\mathbf{V}\in\partial_{B}F(\mathbf{x})\) are nonsingular (regularity on account of B-differential). [PERSON] proved (Lemma 2.6, [15]) that, if \(F\) is BD-regular at \(\mathbf{x}\), then a neighborhood \(N\) of \(\mathbf{x}\) and a constant \(C>0\) exist such that, for any \(\mathbf{y}\in N\) and \(\mathbf{V}\in\partial_{B}F(\mathbf{y})\), \(\mathbf{V}\) is nonsingular and  
\[\left\|\mathbf{V}^{-1}\right\|\leq C.\]  
Throughout this paper, \(\left\|\cdot\right\|\) denotes the 2-norm.  
The notion of semismoothness was originally introduced for functionals by [PERSON] [16]. The following definition is taken from [PERSON] and Sun [17]. A function \(F\) is semismooth at a point \(x\), if \(F\) is locally Lipschitzian at \(\mathbf{x}\) and  
\[\lim_{\mathbf{V}\in\partial F(\mathbf{x}+\mathbf{h}^{\prime}),\mathbf{h}^{\prime} \to\mathbf{h},\mathbf{t}\downarrow 0}\mathbf{V}\mathbf{h}^{\prime}\]  
exists for any \(\mathbf{h}\in R^{n}\). \(F\) is also said semismooth at \(\mathbf{x}\), if it is directionally differentiable at \(\mathbf{x}\) and  
\[\mathbf{V}\mathbf{h}-F^{\prime}\left(\mathbf{x},\mathbf{h}\right)=o\left(\left\|\mathbf{h} \right\|\right).\]  
Scalar products and sums of semismooth functions are still semismooth functions. Piecewise smooth functions and maximum of a finite number of smooth functions are also semismooth. The semismoothness is the almost usually seen assumption on \(F\) in papers dealing with nonsmooth equations because it implies some important properties for convergence analysis of methods in nonsmooth optimization.  
If for any \(\mathbf{V}\in\partial F(\mathbf{x}+\mathbf{h})\), as \(\mathbf{h}\to 0\)  
\[\mathbf{V}\mathbf{h}-F^{\prime}\left(\mathbf{x},\mathbf{h}\right)=O\left(\left\|\mathbf{h}\right\| ^{1+p}\right),\]  
where \(0<\mathrm{p}\leq 1\), then we say \(F\) is p-order semismooth at \(x\). Clearly, p-order semismoothness implies semismoothness. If \(\mathrm{p}=1\), then the function \(F\) is called strongly semismooth. Piecewise \(C^{2}\) functions are examples of strongly semismooth functions.  
[PERSON] and [PERSON] [17] remarked that, if \(F\) is semismooth at \(\mathbf{x}\), then, for any \(\mathbf{h}\to 0\)  
\[F(\mathbf{x}+\mathbf{h})-F(\mathbf{x})-F^{\prime}(\mathbf{x};\mathbf{h})=o\left(\left\|\mathbf{h} \right\|\right),\]  
and, if \(F\) is p-order semismooth at \(\mathbf{x}\), then for any \(\mathbf{h}\to 0\)  
\[F(\mathbf{x}+\mathbf{h})-F(\mathbf{x})-F^{\prime}(\mathbf{x};\mathbf{h})=O\left(\left\|\mathbf{h} \right\|^{1+\mathrm{p}}\right).\]  
**Remark 1**.: _Strong semismoothness of the appropriate function usually implies quadratic convergence of method instead of the superlinear one for semismooth function._  
In turn, [PERSON] and [PERSON] [18] proved that semismoothness of \(F\) at \(\mathbf{x}\) implies that  
\[\sup_{\mathbf{V}\in\partial F(\mathbf{x}+\mathbf{h})}\left\{F(\mathbf{x}+\mathbf{h})-F(\mathbf{x})-\bm {V}\mathbf{h}\right\}=o\left(\left\|\mathbf{h}\right\|\right).\]  
Moreover, if \(F\) is p-order semismooth at \(\mathbf{x}\), then \[\sup_{\mathbf{V}\in\partial F(\mathbf{x}+\mathbf{h})}\left\{F(\mathbf{x}+\mathbf{h})-F(\mathbf{x})-\mathbf{V} \mathbf{h}\right\}=O\left(\left\|\mathbf{h}\right\|^{1+\text{p}}\right).\]