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Solution of monotone complementarity problems with locally Lipschitzian functions

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Abstract

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥ 22 , we extend results which hold for sufficiently smooth functionsF to the nonsmooth case.

In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ 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 top-order semismooth functions. Under a suitable regularity condition and ifF isp-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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  1. Institute for Numerical Mathematics, Technical University of Dresden, D-01062, Dresden, Germany

    Andreas Fischer

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Fischer, A. Solution of monotone complementarity problems with locally Lipschitzian functions. Mathematical Programming 76, 513–532 (1997). https://doi.org/10.1007/BF02614396

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  • DOI: https://doi.org/10.1007/BF02614396

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