Convergence of hybrid steepest-descent methods for variational inequalities.

*(English)*Zbl 1045.49018Summary: Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is $\eta $-strongly monotone and $\kappa $-Lipschitzian on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on $H$. We devise an iterative algorithm which generates a sequence (${x}_{n}$) from an arbitrary initial point ${x}_{0}\in H$. The sequence (${x}_{n}$) is shown to converge in norm to the unique solution ${u}^{*}$ of the variational inequality

$$\langle F\left({u}^{*}\right),v-{u}^{*}\rangle \ge 0,\phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}v\in C\xb7$$

Applications to constrained pseudoinverses are included.

##### MSC:

49J40 | Variational methods including variational inequalities |

47J20 | Inequalities involving nonlinear operators |

90C30 | Nonlinear programming |