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- Solving nonlinear equations by a tabu search strategyPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.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.
- Nonmonotone hybrid tabu search for inequalities and equalities: an experimental studyPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.The main goal of this paper is to analyze the behavior of nonmono- tone hybrid tabu search approaches when solving systems of nonlinear inequalities and equalities through the global optimization of an appro- priate 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 e ort. Two variants of a perturbed pattern search method are implemented as local search. An experimental study involving a variety of problems available in the lit- erature is presented.
- Exploring a multi-objective approach for optimal control problems via a direct multiple shooting methodPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.; Rocha, Ana Maria A. C.; Costa, M. Fernanda P.This paper explores the use of a multi-objective approach through the implementation of a numerical direct multiple shooting (MS) method to solve optimal control problems (OCP). When a direct MS method is used to solve the OCP, a set of `continuity constraints' emerges and should be satisfied together with the other algebraic mixed states and control constraints. To minimize the objective function and satisfy all the constraint conditions, the finite-dimensional optimization problem is reformulated as a multi-objective problem with three objectives to be optimized. An illustrative example is included to show that the present methodology is worth pursuing.
- Combining global tabu search with local search for solving systems of equalities and inequalitiesPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.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 iteration.
- Testing Nelder-Mead Based Repulsion Algorithms for Multiple Roots of Nonlinear Systems via a Two-Level Factorial Design of ExperimentsPublication . Ramadas, Gisela C. V.; Rocha, Ana Maria A. C.; Fernandes, Edite M. G. P.This paper addresses the challenging task of computing multiple roots of a system of nonlinear equations. A repulsion algorithm that invokes the Nelder-Mead (N-M) local search method and uses a penalty-type merit function based on the error function, known as 'erf', is presented. In the N-M algorithm context, different strategies are proposed to enhance the quality of the solutions and improve the overall efficiency. The main goal of this paper is to use a two-level factorial design of experiments to analyze the statistical significance of the observed differences in selected performance criteria produced when testing different strategies in the N-M based repulsion algorithm. The main goal of this paper is to use a two-level factorial design of experiments to analyze the statistical significance of the observed differences in selected performance criteria produced when testing different strategies in the N-M based repulsion algorithm.
- Combined mutation differential evolution to solve systems of nonlinear equationsPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.This paper presents a differential evolution heuristic to compute a solution of a system of nonlinear equations through the global optimization of an appropriate merit function. Three different mutation strategies are combined to generate mutant points. Preliminary numerical results show the effectiveness of the presented heuristic.
- Multiple Roots of Systems of Equations by Repulsion Merit FunctionsPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.; Rocha, Ana Maria A. C.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 minimizer 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 iterations 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.
- On metaheuristics for solving the parameter estimation problem in dynamic systems: A comparative studyPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.; Ramadas, António M. V.; Rocha, Ana Maria A. C.; Costa, M. Fernanda P.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.
- A Multiple Shooting Descent-based Filter Method for Optimal ControlPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.; Rocha, Ana Maria A. C.; Costa, M. Fernanda P.A direct multiple shooting (MS) method is implemented to solve optimal control problems (OCP) in the Mayer form. The use of an MS method gives rise to the so-called ‘continuity conditions’ that must be satisfied together with general algebraic equality and inequality constraints. The resulting finite nonlinear optimization problem is solved by a first-order descent method based on the filter methodology. In the equivalent tri-objective problem, the descent method aims to minimize the objective function, the violation of the ‘continuity conditions’ and the violation of the algebraic constraints simultaneously. The preliminary numerical experiments carried out with a set of benchmark OCP are encouraging.
- Self-adaptive combination of global tabu search and local search for nonlinear equationsPublication . Ramadas, Gisela C. V.; Fernandes, Edite M. G. P.Solving systems of nonlinear equations is a very important task since the problems emerge mostly through the mathematical modelling 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 methods.