demo test functions

The family of Pinter test functions is defined by the following general expression:



Here, s, ak, and fk are integer constants, and Pk are arbitrary n-dimensional polynomials with zero intercept. Thus, by design, the global minimum of f(x) is zero and it is located at x = x*

When n = 1, the expression above defines the family of one-dimensional Pinter functions. A typical plot of a function in this family is included below. The right-hand side is merely a zoomed version of the same function, expanded around x = 300. The plots illustrate visually the large amount of unstructured local minima, and thus the inherent difficulty that this function poses for finding its global minimum. This difficulty is what makes the Pinter functions good candidates to conduct solver performance tests.

The difficulty of the function can be controlled by tuning the function parameters. For example, the function would be more difficult if the ratio between s and the ak’s is small, and the frequencies fk are large. For one-dimensional functions, a value of kmax = 2 suffices to make the functions difficult enough. It can be proven that, for every solver V, and for any time value t, it is always possible to design a difficult one-dimensional Pinter function, such that V will not be able to find its global minimum in a time less than t.