This paper deals with the monotone homotopy methods for solving the sy
stem of nonlinear equations. In Sect. 2 the homotopy with a distance-m
onotone homotopy path is discussed and it is proved that a homotopy wi
th a given structure has a distance monotone homotopy path under some
regular conditions. Then two structure-variable homotopy algorithms ca
lled local straighten-up method and global straighten-up method are de
veloped to approximate the optimal homotopy, and an adaptive step-size
control strategy for these algorithms is proposed respectively. A new
method for analyzing the convergence is presented, which is based on
the geometrical properties of the algorithm. Then the convergence of t
he algorithm is proved under certain regular conditions. Finally two n
umerical examples are given to illustrate the effectiveness of the abo
ve two algorithms.