Nine-Point Iterated Rectangle Dichotomy for Finding All Local Minima of Unknown Bounded Surface

Abstract

Abstract A method is suggested to find all local minima and the global minimum of an unknown two-variable function bounded on a given rectangle regardless of the rectangle area. The method has eight inputs: five inputs defined straightforwardly and three inputs, which are adjustable. The endpoints of the initial intervals constituting the rectangle and a formula for evaluating the two-variable function at any point of this rectangle are the straightforward inputs. The three adjustable inputs are a tolerance with the minimal and maximal numbers of subintervals along each dimension. The tolerance is the secondary adjustable input. Having broken the initial rectangle into a set of subrectangles, the nine-point iterated rectangle dichotomy “gropes” around every local minimum by successively cutting off 75 % of the subrectangle area or dividing the subrectangle in four. A range of subrectangle sets defined by the minimal and maximal numbers of subintervals along each dimension is covered by running the nine-point rectangle dichotomy on every set of subrectangles. As a set of values of currently found local minima points changes no more than by the tolerance, the set of local minimum points and the respective set of minimum values of the surface are returned. The presented approach is applicable to whichever task of finding local extrema is. If primarily the purpose is to find all local maxima or the global maximum of the two-variable function, the presented approach is applied to the function taken with the negative sign. The presented approach is a significant and important contribution to the field of numerical estimation and approximate analysis. Although the method does not assure obtaining all local minima (or maxima) for any two-variable function, setting appropriate minimal and maximal numbers of subintervals makes missing some minima (or maxima) very unlikely.