  In this paper, the problem of safe global maximization (it should not be
confused with robust optimization) of expensive noisy black-box functions
satisfying the Lipschitz condition is considered. The notion "safe" means that
the objective function $f(x)$ during optimization should not violate a "safety"
threshold, for instance, a certain a priori given value $h$ in a maximization
problem. Thus, any new function evaluation (possibly corrupted by noise) must
be performed at "safe points" only, namely, at points $y$ for which it is known
that the objective function $f(y) > h$. The main difficulty here consists in
the fact that the used optimization algorithm should ensure that the safety
constraint will be satisfied at a point $y$ before evaluation of $f(y)$ will be
executed. Thus, it is required both to determine the safe region $\Omega$
within the search domain~$D$ and to find the global maximum within $\Omega$. An
additional difficulty consists in the fact that these problems should be solved
in the presence of the noise. This paper starts with a theoretical study of the
problem and it is shown that even though the objective function $f(x)$
satisfies the Lipschitz condition, traditional Lipschitz minorants and
majorants cannot be used due to the presence of the noise. Then, a
$\delta$-Lipschitz framework and two algorithms using it are proposed to solve
the safe global maximization problem. The first method determines the safe area
within the search domain and the second one executes the global maximization
over the found safe region. For both methods a number of theoretical results
related to their functioning and convergence is established. Finally, numerical
experiments confirming the reliability of the proposed procedures are
performed.
