  This paper deals with two kinds of the one-dimensional global optimization
problems over a closed finite interval: (i) the objective function $f(x)$
satisfies the Lipschitz condition with a constant $L$; (ii) the first
derivative of $f(x)$ satisfies the Lipschitz condition with a constant $M$. In
the paper, six algorithms are presented for the case (i) and six algorithms for
the case (ii). In both cases, auxiliary functions are constructed and
adaptively improved during the search. In the case (i), piece-wise linear
functions are constructed and in the case (ii) smooth piece-wise quadratic
functions are used. The constants $L$ and $M$ either are taken as values known
a priori or are dynamically estimated during the search. A recent technique
that adaptively estimates the local Lipschitz constants over different zones of
the search region is used to accelerate the search. A new technique called the
\emph{local improvement} is introduced in order to accelerate the search in
both cases (i) and (ii). The algorithms are described in a unique framework,
their properties are studied from a general viewpoint, and convergence
conditions of the proposed algorithms are given. Numerical experiments executed
on 120 test problems taken from the literature show quite a promising
performance of the new accelerating techniques.
