Searching minima of an N-dimensional surface: A robust valley following method

Abstract

A procedure is proposed to follow the “minimum path” of a hypersurface starting anywhere in the catchment region of the corresponding minimum. The method uses a modification of the so-called “following the reduced gradient” [1]. The original method connects points where the gradient has a constant direction. In the present letter, this is replaced by the successive directions of the tangent of the searched curve. The resulting pathway is that valley floor gradient extremal which belongs to the smallest (absolute) eigenvalue of the Hessian. The new method avoids third derivatives of the objective function. The effectiveness of the algorithm is demonstrated by using a polynomial test, the notorious Rosenbrock function in two, 20, and in 100 dimensions.