A universal algorithm is presented in the paper which makes it possible to find out whether a given relative extremum of a scalar-valued function of a vector argument is also the absolute one and, if not, to find a point at which a relative extremum-seeking procedure can be restarted in order that a better relative extremum—i.e. one at which the given function takes a less, alternatively greater, value—may be obtained. Repeated application of the described procedure in combination with an arbitrary relative extremum-seeking procedure leads to determining the absolute extremum. The possibility of application to stochastic and integer programming problems is discussed.