Multiple roots of systems of equations by repulsion merit functions

In this paper we address the problem of computing multiple roots of a system of nonlinear equations through the global optimization of an appropriate merit function. The search procedure for a global min- imizer of the merit function is carried out by a metaheuristic, known as harmony search, which does not require any derivative information. The multiple roots of the system are sequentially determined along several ite- rations of a single run, where the merit function is accordingly modified by penalty terms that aim to create repulsion areas around previously computed minimizers. A repulsion algorithm based on a multiplicative kind penalty function is proposed. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.Fundação para a Ciência e a Tecnologia (FCT

Constrained dogleg methods for nonlinear systems with simple bounds

We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problem

GenAnneal: Genetically modified Simulated Annealing

A modification of the standard Simulated Annealing (SA) algorithm is presented for finding the global minimum of a continuous multidimensional, multimodal function. We report results of computational experiments with a set of test functions and we compare to methods of similar structure. The accompanying software accepts objective functions coded both in Fortran 77 and C++. Program summary: Title of program:GenAnneal. Catalogue identifier:ADXI_v1_0. Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXI_v1_0. Program available from: CPC Program Library, Queen's University of Belfast, N. Ireland. Computer for which the program is designed and others on which it has been tested: The tool is designed to be portable in all systems running the GNU C++ compiler. Installation: University of Ioannina, Greece on Linux based machines. 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 bytes in distributed program, including test data, etc.:84 885. No. of lines in distributed program, including test data, etc.:14 896. Distribution format: tar.gz. Nature of 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 are frequently trapped in local minima. Global optimization is hence 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). Typical running time: Depending on the objective function. Method of solution: We modified the process of step selection that the traditional Simulated Annealing employs and instead we used a global technique based on grammatical evolution. © 2006 Elsevier B.V. All rights reserved

An expedition to technical secrets of clarinet playing

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