Limits and improvements on searching and optimization: from one dimensional problems to multi-objective optimization

• Autor(a) principal: Ivo Fagundes David de Oliveira
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/41216; https://orcid.org/0000-0001-8450-5054
• topic: Binary searching; Root searching; Line searching; List searching; Gradient method; Multiobjective optimization; Backtracking; Engenharia elétrica; Otimização matemática; Otimização multiobjetivo

This thesis presents a series of improvements on four different classical searching methods employed for solving different well established problems. The methods improved on and their corresponding problems are: (i) the bisection method for continuous root-searching problems; (ii) the binary search algorithm for discrete list-searching; (iii) the back-tracking technique for inexact Armijo-type searching; and (iv) the n-dimensional steepest descent method for non-linear multi-objective optimization. Different types of improvements are aimed for in each context that produce an overall reduction in the the number of calls to the external function being searched. However, all four improvements proposed have one thing in common: the worst-case upper-bound of our methods either outperform the state-of-the-art, or, where the state-of-the-art has already attained an optimal worst-case performance, we match the performance of the optimal bound while improving on either average performance, asymptotic performance or both. Thus, in this sense, the methods we propose are \emph{strict} improvements on classical solutions, attained with no additional assumptions on the problems considered nor with any additional costs other than the computation of the methods themselves. The manuscript starts with a broad introduction which discusses the importance of the problems considered and the classical solutions employed in several different fields. The main contributions are given in the following four chapters; one corresponding to each problem tackled. Each chapter corresponds to one published (or soon to be published) result intimately related to the four problems considered which are augmented with original unpublished material. Finally, in the sixth and final chapter we point to possible ramifications of the findings hereby delineated which present potential for future developments.