Reducing transformation and global optimization

In this paper, we give new results on the Alienor method of dimension reduction. This technique is used to solve multidimensional global optimization problems of type minx∈X f(x) where f is a non convex Lipschitz function and X   a compact set of Rn(n⩾2) defined by Lipschitz constraints. The idea is to construct an α-dense curve h in the feasible set X. The global minimum of f on X is then approximated by the global minimum of f on the curve h. That is, our problem has become a one-dimensional problem which can be solved by the Piyavskii–Shubert method. Examples of these curves and numerical implementations on several test functions are given.