Solution of monotone complementarity problems with locally Lipschitzian functions.

*(English)*Zbl 0871.90097Summary: 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 \(1+p\).

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) |

##### Keywords:

locally Lipschitzian function; monotone function; descent method; complementarity problems; generalized Newton method; semismooth functions
Full Text:
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