An Algorithm for Monotone Complementarity Problems with Locally Lipschitzian Functions

Abstract

We consider the complementarity problem CP(F) of finding x ∈ IRⁿ such that

$$ F(x) \geqslant 0,\;x \geqslant 0,\;{x^T}F(x) = 0 $$

where F: IRⁿ → IRⁿ is a given monotone locally Lipschitzian function. From [11] and many subsequently published papers it is well-known that this problem can be transformed into an equivalent nonlinear (and possibly nonsmooth) system of equations

$$ \Phi (x): = \left[ {\begin{array}{*{20}{c}} {\phi \left( {{x_1},\,{F_1}(x)} \right)} \\ \vdots \\ {\phi \left( {{x_n},\,{F_n}(x)} \right)} \\ \end{array} } \right] = 0 $$

provided that the function Φ: IR² → IR satisfies

$$ \phi \left( {a,b} \right):0 \Leftrightarrow a \geqslant 0,\,b \geqslant 0,\,ab = 0\,\,\,\,\left( {a,b \in \mathbb{R}} \right) $$

In this paper the special function

$$ \phi \left( {a,b} \right): = \sqrt {{{a^2} + {b^2}}} - a - b $$

will be employed, see [3]. Recalling the results obtained by different authors up to now (see, for instance, [2, ?, 5, ?, 8, 9]), this is a useful approach for the solution of complementarity problems CP(F) if the underlying function F is sufficiently smooth (F ∈ C ^l or stronger assumptions). Based on the papers just cited and on the recent report [6] we will suggest an algorithm for the solution of complementarity problems CP(F), where F is a monotone but only locally Lipschitzian function. Moreover, global and local convergence properties of this algorithm will be stated.