Relatórios de Pesquisa

4/2011 Approximation and Quantiles of the Distribution of the Modified Likelihood Ratio Criteria for Covariance Matrix Hypothesis Testing andMonitoring
Mario A. Gneri, Emanuel P. Barbosa, Ariane Meneguetti

Sugiura (1969) gives an asymptotic expansion of the modified likelihood ratio criteria for testing the hypothesis that a covariance matrix is equal to a given matrix. An improvement of this expansion is presented here. Numerical comparisons via simulation with the original Sugiura's approximation to the distribution of the criteria confirm the superiority of our expansion. This enable us to use the proposed method in usual hypotheses testing and in applications where extreme tail quantiles are necessary, as for instance, for monitoring dispersion in multivariate processes quality control charts.

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3/2011 Sistemas não lineares via região de confiança: o algoritmo de Levenberg-Marquardt
John Lenon C. Gardenghi, Sandra A. Santos

This work consists in the study and the computational implementation of the Levenberg-Marquardt algorithm, as proposed by Moré (1978), for the solution of nonlinear systems by means of an unconstrained optimization problem. Such a method is globally convergent and can be implemented in an efficient and robust way. Our goal in the preparation of this text was to follow the analysis presented by Moré, including further details and the main ideas of trust-region methods. Moreover, we aimed to organize the necessary concepts to follow the details concerning the numerical andcomputational implementation in a concise but thorough way. The numerical experiments validated the implementation and were totally coded using the CAS Maxima, not only the main program, but also the whole set of functions and routines involving the computational linear algebra of the algorithm.

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2/2011 Otimização e Análise Convexa: Aspectos Teóricos e Aplicações
Matheus Souza, Maria A. Diniz-Ehrhardt

Convex Optimization is a special class of problems in mathematical optimization where the cost function and the feasible set are both convex. There are several effcient and trustable methods, mainly interior-point methods, to solve problems like these. This is due to the well developed theory they are based on and to their frequent occurrence in practical applications. Classical problems of convex optimization and their applications are to be analysed.

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1/2011 Nonlinear Elastodynamics with Radial Symmetry, Model and Qualitative Investigation
Artur Lewis Gower

This article deduces a model, stated as an integral equation, for any nonlinear elastic isotropic material undergoing a radially symmetric deformation. Such a model is useful in the study of an explosion, or a spherically symmetric impact. Determining the effects of nonlinear wave propagation, in relation to linear propagation, can be truly challenging in 3D dimensions. By reducing the system to a 1D radial partial integral equation numerical simulations are more accurate and manageable. Also, understanding the radially symmetric model sheds light on the qualitative behaviour of the full 3D nonlinear system. An emphasis is given on an intuitive understanding of the dynamics. After deducing the general integral model we present discontinuous jump conditions, and then discuss and substitute the Mooney-Rivlin approximation for the material. We point out how the model for the linearised material can approximate a Mooney-Rivlin material, and subsequentially present the analytical solution to some important cases of the linearised material. The appendix attempts to be a rather complete exposition which departs from first principles, where the theoretical basis follows the axiomatic treatment of elasticity and the integral formulation of balance principles.

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18/2010 Description of Some Ground States by Puiseux Technics
Eduardo Garibaldi, Philippe Thieullen
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17/2010 Programação Quadrática Sequencial e Condições de Qualificação
Fernanda Teles Nunes, Maria A. Diniz-Ehrhardt

In the context of constrained optimization problems, we face the optimality conditions and also constraint qualification. Our aim is to study with details several constraint qualifications, highlighting the constant positive linear dependence condition, and its influence in Sequential Quadratic Programming algorithms convergence. The relevance of this study is in the fact that convergence results having as hypothesis weak constraint qualifications are stronger than those based on stronger constraint qualifications. Numerical experiments will be done with the purpose of investigating the efficiency of these methods to solve problems with different constraint qualifications and to compare two diferent kinds of line search, monotone and nonmonotone. We want to confirm the hypothesis that algorithms based on a nonmonotone line search act against the Maratos Effect, very common while solving minimization problems through Sequential Quadratic Programming methods.

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16/2010 Hessian Matrices via Automatic Differentiation
Robert Mansel Gower, Margarida P. Mello

We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther's state transformations, synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebraic, lead to a new framework for Hessian computation. This is illustrated by giving a new correctness proof for Griewank and Walther's reverse Hessian algorithm and by developing edge pushing, a new truly reverse Hessian computation algorithm that fully exploits the Hessian's symmetry. Computational experiments compare the performance of edge pushing on sixteen functions from the CUTE collection against two algorithms available as drivers of the software ADOL-C, and the results are very promising.

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15/2010 Globally convergent modifications to the Method of Moving Asymptotes and the solution of the subproblems using trust regions: theoretical and numerical results
Márcia A. Gomes-Ruggiero, Mael Sachine, Sandra A. Santos

An alternative strategy to solve the subproblems of the Method of Moving Asymptotes (MMA) is presented, based on a trust-region scheme applied to the dual of the MMA subproblem. At each iteration, the objective function of the dual problem is approximated by a regularized spectral model. A globally convergent modification to the MMA is also suggested, in which the conservative condition is relaxed by means of a summable controlled forcing sequence. Another modification to the MMA previously proposed by the authors [\emph{Optim. Methods Softw.}, 25 (2010), pp. 883--893] is recalled to be used in the numerical tests. This modification is based on the spectral parameter for updating the MMA models, so as to improve their quality. The performed numerical experiments confirm the efficiency of the indicated modifications, especially when jointly combined. This report contains all the global convergence results and the complete set of numerical and graphical elements that sustain our performance analysis.

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14/2010 The effective potential and transshipment in thermodynamic formalism at temperature zero
Eduardo Garibaldi, Artur O. Lopes
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13/2010 An Improved |S| Control Chart for Multivariate Process Variability Monitoring based on Cornish-Fisher Correction
Emanuel P. Barbosa, Mario A. Gneri, Ariane Meneguetti

This paper presents an improved version of the generalized variance |S| control chart for multivariate process dispersion monitoring, based on the Cornish-Fisher for-mula for non-normality correction of the usual normal based 3-sigma limits chart. The exact sample distribution of jSj doesn't have a simple known form for dimension p > 2, and we show here that the information from its 3r.d and 4t.h order moments or cumulants are sufficient for a satisfactory approximation. The performance of this corrected control chart is compared (in terms of false alarm risk) with the original normal based chart and the exact distribution based chart (for p = 2 and p = 3) where in the last case (p = 3), the exact distribution is obtained by simulation methods. This study shows that the control limits corrections do remove the drawback of excess of false alarm associated with the traditional normal based |S| control chart. Finally, the proposed new chart is illustrated with a numerical example of application with real data.

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