Uma introdução à otimização contínua: estudo dos métodos de Polak-Ribiere e de Newton truncado para a minimização de funções

Abstract

In this monograph we deal with the theme of continuous optimization, making a study of the order one, Polak-Ribière, and order two, Newton truncated methods. This work sought to understand the advantages of each of the two methods when used to minimize some types of multidimensional functions present in the optimization literature, as well as providing a not so formal introduction to the subject of optimization, more specifically the case of minimization. Firstly, a bibliographic study of the fundamentals of continuous optimization was conducted, aiming to review the mathematical tools necessary to understand the proposed theme. Then, another detailed bibliographic study was conducted on the main algorithmic and mathematical aspects of the two methods mentioned above. Finally, a test was conducted where we simulated the minimization of the same 5 functions for both methods in order to compare their performance, using the implementation of these methods offered by the Scipy library of the Python programming language. The metrics used for this comparison, which are offered at the end of the iteration by the Scipy library, were number of function tests, method iteration number, number of Jacobian tests. The results pointed to a better performance of Newton's Truncated method in all cases of the proposed functions, but the Polak-Ribière method was also very fast in delivering the solution. We conclude with this work that the subject of optimization, as well as the two specific methods discussed here, are a good introduction for those interested in optimization.