The paper deals with complementarity problems CP(
F), where the underlying function
F is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(
F) as a system of equations
φ(
x)=0 or as the problem of minimizing the merit function
Θ=1/2∥
Φ∥
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 ε
δθ(
x) is necessary and sufficient for
x to be a solution of CP(
F). Moreover, for monotone functions
F, a simple derivative-free algorithm that reduces
Θ 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.
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