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

A procedure is proposed to follow the "minimum path" of a hypersurface star ting anywhere in the catchment region of the corresponding minimum. The met hod uses a modification of the so-called "following the reduced gradient" [ 1]. The original method connects points where the gradient has a constant d irection. In the present letter, this is replaced by the successive directi ons of the tangent of the searched curve. The resulting pathway is that val ley floor gradient extremal which belongs to the smallest (absolute) eigenv alue of the Hessian. The new method avoids third derivatives of the objecti ve function. The effectiveness of the algorithm is demonstrated by using a polynomial test, the notorious Rosenbrock function in two, 20, and in 100 d imensions. (C) 2001 Elsevier Science Ltd. All rights reserved.