In this paper, Lipschitz univariate constrained global optimization problems
where both the objective function and constraints can be multiextremal are
considered. The constrained problem is reduced to a discontinuous unconstrained
problem by the index scheme without introducing additional parameters or
variables. A Branch-and-Bound method that does not use derivatives for solving
the reduced problem is proposed. The method either determines the infeasibility
of the original problem or finds lower and upper bounds for the global
solution. Not all the constraints are evaluated during every iteration of the
algorithm, providing a significant acceleration of the search. Convergence
conditions of the new method are established. Test problems and extensive
numerical experiments are presented.