Local analysis of Newton-type methods for variational inequalities and nonlinear programming (Q1317308)

This paper mainly considers the Newton-type methods for variational inequalities as follows: \[ \langle\varphi(z), y- z\rangle\geq 0,\quad\forall y\in K,\tag{1} \] where \(\varphi\) is a continuously differentiable mapping from \(\mathbb{R}^ q\) into \(\mathbb{R}^ q\), \(K\subset \mathbb{R}^ q\) is a closed convex subset. (1) is equivalent to \[ \varphi(z)+ N(z)\ni 0,\tag{2} \] where \(N(z)\) is the normal cone to \(K\) at to \(z\in K\). A solution \(\bar z\) of (2) is said to be semistable if \(c_ 1>0\) and \(c_ 2>0\) exist such that, for all \((z,\delta)\in \mathbb{R}^ q\times \mathbb{R}^ q\), solution of \(\varphi(z)+ N(z)\ni \delta\), and \(\| z- \bar z\|\leq c_ 1\), then \(\| z- \bar z\|\leq c_ 2\| \delta\|\). The author shows that this condition ensures the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth functions is given. For applying the results to nonlinear programming problems, the second part of the paper considers some particular case. This allows to strengthen the known results, the main point being a characterization of the superlinear convergence of \(\{x^ k\}\) assuming a weak second-order condition without strict complementarity.