An interactive global optimization algorithm for geological problems

A number of problems in geology can be formulated so that they consist of optimizing a real-valued function (termed the objective function) on some interval or over some region. Many methods are available for solution if the function is unimodal within the domain of interest. Direct methods, involving only function evaluations, are particularly useful in geological problems where the objective function may be strongly nonlinear and constructed from sampled data. In practical problems, the objective function often is not unimodal. Standard optimization routines are not capable of distinguishing between local extrema or of locating the global extremum, which is the point of interest in most cases. The usual approach—trying several different starting points in the hope that the best local extremum found is the global extremum—is inefficient and unreliable. An ancillary algorithm has been developed which avoids these problems and which couples with a variety of local optimization routines. The algorithm first constructs a grid of objective function values over some feasible region. The region dimensions and grid spacings are based on specific problem considerations. First differences are then calculated for successive points along each grid line and monitored in sign only, which rapidly locates extrema. User interaction determines how many of these extrema will undergo further investigation, which is carried out by passing locations to a local optimization subroutine. The algorithm has proved successful on a number of problems. A geological example—determination of benthic mixing parameters in deep-sea sediments via minimization of stratigraphic offset between ¹⁸ O signals from two different species of planktonic foraminifera—is given. FORTRAN code is provided for the global optimization routine, a golden section search subroutine for one-dimensional objective functions, and a simplex subroutine for multidimensional problems.