Jacobi iterative method for a class of complementarity problems

Abstract

This thesis studies the Complementarity Problem (CP) with special structure where the given mapping f of the CP is differentiable, strictly diagonally isotone and off-diagonally isotone on R$^n$. This type of problem can be found in the more realistic traffic equilibrium problem with elastic demands. Existence of a solution to this specific CP is guaranteed by strong copositivity and continuity of a mapping f. This thesis presents a simple yet practically useful Jacobi-type iterative solution algorithm for this specific CP, and partially obtains its convergence properties. Its main properties are as follows; First, the sequence of even number iterates generated by the suggested algorithm, i.e., {Z$^{2k}$} converges to a lower bound z$^L$ of all solutions for this specific CP. Second, the sequence of odd number iterates, i.e., {Z$^{2k+1}$} converges to an upper bound z$^U$. Third, all solutions of this specific CP are contained in the order interval $<z^L,\; z^U>$ This study also investigates the convergence conditions for the linear CP with this specific structure.