  A global optimization problem is studied where the objective function $f(x)$
is a multidimensional black-box function and its gradient $f'(x)$ satisfies the
Lipschitz condition over a hyperinterval with an unknown Lipschitz constant
$K$. Different methods for solving this problem by using an a priori given
estimate of $K$, its adaptive estimates, and adaptive estimates of local
Lipschitz constants are known in the literature. Recently, the authors have
proposed a one-dimensional algorithm working with multiple estimates of the
Lipschitz constant for $f'(x)$ (the existence of such an algorithm was a
challenge for 15 years). In this paper, a new multidimensional geometric method
evolving the ideas of this one-dimensional scheme and using an efficient
one-point-based partitioning strategy is proposed. Numerical experiments
executed on 800 multidimensional test functions demonstrate quite a promising
performance in comparison with popular DIRECT-based methods.
