  The problem is analyzed of extrapolating power series, derived for an
asymptotically small variable, to the region of finite values of this variable.
The consideration is based on the self-similar approximation theory. A new
method is suggested for defining the odd self-similar factor approximants by
employing an optimization procedure. The method is illustrated by several
examples having the mathematical structure typical of the problems in
statistical and chemical physics. It is shown that the suggested method
provides a good accuracy even when the number of terms in the perturbative
power series is small.
