Solving nonlinear equations by a tabu search strategy

Solving systems of nonlinear equations is a problem of particular importance since they emerge through the mathematical modeling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a metaheuristic, called Directed Tabu Search (DTS) [16], is able to converge to the solutions of a set of problems for which the fsolve function of MATLAB® failed to converge. We also show the effect of the dimension of the problem in the performance of the DTS.Fundação para a Ciência e a Tecnologia (FCT

Self-adaptive combination of global tabu search and local search for nonlinear equations

Solving systems of nonlinear equations is a very important task since the problems emerge mostly through the mathematical modeling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a self-adaptive combination of a metaheuristic with a classical local search method is able to converge to some difficult problems that are not solved by Newton-type methodsFundação para a Ciência e a Tecnologia (FCT

Solving systems of inequalities and equalities by a nonmonotone hybrid tabu search method

Series : AIP Conference Proceedings, Vol. 1479This paper presents a derivative-free nonmonotone hybrid tabu search to compute a solution of overdetermined systems of inequalities and equalities through the global optimization of an appropriate merit function. The proposed algorithm combines global and local searches aiming to reduce computational effort. Preliminary numerical results show the effectiveness of the combined heuristic.Fundação para a Ciência e a Tecnologia (FCT

Combining global tabu search with local search for solving systems of equalities and inequalities

This papers aims at providing a combined strategy for solving systems of equalities and inequalities. The combined strategy uses two types of steps: a global search step and a local search step. The global step relies on a tabu search heuristic and the local step uses a deterministic search known as Hooke and Jeeves. The choice of step, at each iteration, is based on the level of reduction of the l2-norm of the error function observed in the equivalent system of equations, compared with the previous iterationFundação para a Ciência e a Tecnologia (FCT

Nonmonotone hybrid tabu search for Inequalities and equalities: an experimental study

The main goal of this paper is to analyze the behavior of nonmonotone hybrid tabu search approaches when solving systems of nonlinear inequalities and equalities through the global optimization of an appropriate merit function. The algorithm combines global and local searches and uses a nonmonotone reduction of the merit function to choose the local search. Relaxing the condition aims to call the local search more often and reduces the overall computational effort. Two variants of a perturbed pattern search method are implemented as local search. An experimental study involving a variety of problems available in the literature is presented.Fundação para a Ciência e a Tecnologia (FCT

Simplified tabu search with random-based searches for bound constrained global optimization

This paper proposes a simplified version of the tabu search algorithm that solely uses randomly generated direction vectors in the exploration and intensification search procedures, in order to define a set of trial points while searching in the neighborhood of a given point. In the diversification procedure, points that are inside any already visited region with a relative small visited frequency may be accepted, apart from those that are outside the visited regions. The produced numerical results show the robustness of the proposed method. Its efficiency when compared to other known metaheuristics available in the literature is encouraging.FCT - Fundação para a Ciência e a Tecnologia(UIDB/00013/2020); FCT – Fundação para a Ciência e Tecnologia within the R&D Units Project Scope: UIDB/00319/2020, UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM

On metaheuristics for solving the parameter estimation problem in dynamic systems: A comparative study

This paper presents an experimental study that aims to compare the practical performance of well-known metaheuristics for solving the parameter estimation problem in a dynamic systems context. The metaheuristics produce good quality approximations to the global solution of a finite small-dimensional nonlinear programming problem that emerges from the application of the sequential numerical direct method to the parameter estimation problem. Using statistical hypotheses testing, significant differences in the performance of the metaheuristics, in terms of the average objective function values and average CPU time, are determined. Furthermore, the best obtained solutions are graphically compared in relative terms by means of the performance profiles. The numerical comparisons with other results in the literature show that the tested metaheuristics are effective in achieving good quality solutions with a reduced computational effort.The authors would like to acknowledge the financial support of CIDEM, R&D Unit, funded by the Portuguese Foundation for the Development of Science and Technology (FCT), Ministry of Science, Technology and Higher Education, under the Project UID/EMS/0615/2016, and of COMPETE: POCI-01-0145-FEDER-007043 and FCT within the Projects UID/CEC/00319/2013 and UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance