An Interior Ellipsoid Algorithm for Fixed Points

We consider the problem of approximating xed points of non smooth con tractive functions with using of the absolute error criterion In we proved that the upper bound on the number of function evaluations to compute approximations is O n ln ln q ln n in the worst case where q is the contraction factor and n is the dimension of the problem This upper bound is achieved by the circumscribed ellipsoid CE algorithm combined with a dimensional de ation process In this paper we present an inscribed ellipsoid IE algorithm that enjoys O n ln ln q ln n bound Therefore the IE algorithm has almost the same modulo multiplicative constant number of function evaluations as the nonconstructive centroid method We conjecture that this bound is the best possible for mildly contractive functions q in moderate dimensional case A rmative solution of this conjecture would imply that the IE algorithm and the centroid algorithms are almost optimal in the worst case In particular they are much faster than the simple iteration method that requires l ln ln q m function evaluations to solve the problem