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. 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. Program summary

Multilocal Programming and Applications

“…These techniques aim at avoiding repetitive identification of the same solutions. Well-known examples are the clustering methods [50,51,52]. Other techniques that aim to escape from previously computed solutions, in general local solutions, are based on constructing auxiliary functions via a current local solution of the original problem [55,63,64].…”

A Derivative-Free Filter Driven Multistart Technique for Global Optimization

“…One of the most known stochastic algorithms is the multistart. In the last decade some research has been focused on this type of methods [1,6,[9][10][11]; see also [7] and the references therein included. The underlying idea of this method is to sample uniformly a point from the search region and to perform a local search, starting from this point, to obtain an optimal (local) solution, using a local technique.…”

Multilocal Programming: A Derivative-Free Filter Multistart Algorithm

“…Sampled points from these prohibited regions are discarded since the local search procedure would converge most certainly to an already located solution. MinFinder is an example of a clustering algorithm that competes with multistart when global and some local minimizers are required [21,22]. Alternatively, niching, deflecting and stretching techniques may be combined with global optimization methods, like the simulated annealing, evolutionary algorithm and the particle swarm optimization, to discover the global and some specific local minimizers of a problem [17,18,20].…”

“…The goal of most multistart methods presented in the literature is to locate multiple solutions of bound constrained optimization problems [1,21,23,24] (see also [15] and the references therein included). Multistart may also be used to explore the search space and converge to a global solution of nonlinear optimization problems [6].…”