MinFinder: Locating all the local minima of a function

Abstract

A new stochastic clustering algorithm is introduced that aims to locate all the local minima of a multidimensional continuous and differentiable function inside a bounded domain. The accompanying software (MinFinder) is written in ANSI C++. However, the user may code his objective function either in C++, C or Fortran 77. We compare the performance of this new method to the performance of Multistart and Topographical Multilevel Single Linkage Clustering on a set of benchmark problems. Program summary Title of program:MinFinder Catalogue identifier:ADWU Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWU Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computer for which the program is designed and others on which is has been tested:The tool is designed to be portable in all systems running the GNU C++ compiler Installation:University of Ioannina, Greece Programming language used:GNU-C++, GNU-C, GNU Fortran 77 Memory required to execute with typical data:200 KB No. of bits in a word:32 No. of processors used:1 Has the code been vectorized or parallelized?:no No. of lines in distributed program, including test data, etc.:5797 No. of bytes in distributed program, including test data, etc.:588 121 Distribution format:gzipped tar file Nature of the physical problem:A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques can be trapped in any local minimum. Global optimization is then the appropriate tool. For example, solving a non-linear system of equations via optimization, employing a “least squares” type of objective, one may encounter many local minima that do not correspond to solutions, i.e. they are far from zero. Method of solution:Using a uniform pdf, points are sampled from the rectangular search domain. A clustering technique, based on a typical distance and a gradient criterion, is used to decide from which points a local search should be started. The employed local procedure is a BFGS version due to Powell. Further searching is terminated when all the local minima inside the search domain are thought to be found. This is accomplished via the double-box rule. Typical running time:Depending on the objective function