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Solution of monotone complementarity problems with locally Lipschitzian functions. (English) Zbl 0871.90097
Summary: The paper deals with complementarity problems $$\text{CP}(F)$$, where the underlying function $$F$$ is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of $$\text{CP}(F)$$ as a system of equations $$\Phi(x)=0$$ or as the problem of minimizing the merit function $$\Psi=\frac{1}{2}|\Phi|_2^2$$, we extend results which hold for sufficiently smooth functions $$F$$ to the nonsmooth case.
In particular, if $$F$$ is monotone in a neighbourhood of $$x$$, it is proved that $$0\in\partial\Psi(x)$$ is necessary and sufficient for $$x$$ to be a solution of $$\text{CP}(F)$$. Moreover, for monotone functions $$F$$, a simple derivative-free algorithm that reduces $$\Psi$$ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended to $$p$$-order semismooth functions. Under a suitable regularity condition and if $$F$$ is $$p$$-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of $$1+p$$.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
##### Software:
PATH Solver; QPCOMP
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