(8) OPTIMIZATION: Shubert-Piyavskii Method to Find Global Minimum

Global Minimum of Univariate Function over an Interval [a, b]

The Shubert-Piyavskii method differs from the previously discussed methods in this series because it considers the whole interval [a, b] instead of focusing on specific points. As a result, this method can converge to the global minimum of the function, regardless of whether there are any local minimums or if the function is unimodal.

However, to apply the Shubert-Piyavskii method it is necessary for the function to be Lipschitz continuous. This means that the function must be continuous and that there exists a maximum limit for the size of its derivative.

Confusing? Let´s break it!

As previously explained, for a function f to be Lipschitz continuous over an interval [a, b], f must be continuous in the interval [a, b], and the size of the derivative of f has a maximum limit. Remember that the derivative of f corresponds to the rate of change of f. So, if the derivative of f has a maximum limit value, the rate of change of f has a maximum value too.

In simple terms, Lipschitz continuity describes a function that has a certain grade of ‘smoothness’ property. A function is Lipschitz continuous if there exists a constant number, called…