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Master's Dissertation
DOI
10.11606/D.45.2018.tde-11052018-113001
Document
Author
Full name
Leonardo Makoto Mito
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Haeser, Gabriel (President)
Passos, Marcelo Dias
Senne, Thadeu Alves
Title in Portuguese
O problema de cobertura via geometria algébrica convexa
Keywords in Portuguese
Geometria algébrica real
Problema de cobertura
Programação semidefinida
Restauração inexata
Abstract in Portuguese
Este trabalho é focado num problema clássico das Ciências e Engenharia, que consiste em cobrir um objeto por esferas de mesmo raio, a ser minimizado. A abordagem prática usual conta com sérias desvantagens. Logo, faz-se necessário trabalhar com isto de forma diferenciada. A técnica proposta aqui envolve a utilização de resultados célebres da geometria algébrica real, que tem como peça central o positivstellensatz de Stengle e, fazendo a devida relação entre esses resultados e otimização com restrições envolvendo representações naturais por somas de quadrados, é possível reduzir o problema original a um de programação semidefinida não linear. Mas, por contar com particularidades que favorecem a aplicação do paradigma de restauração inexata, esta foi a técnica utilizada para resolvê-lo. A versatilidade da técnica e a possibilidade de generalização direta dos objetos envolvidos destacam-se como grandes vantagens desta abordagem, além da visão algébrica inovadora do problema.
Title in English
The covering problem via convex algebraic geometry
Keywords in English
Covering problem
Inexact restoration
Real algebraic geometry
Semidefinite programming
Abstract in English
This work is focused on a classic problem from Engineering. Basically, it consists of finding the optimal positioning and radius of a set of equal spheres in order to cover a given object. The common approach to this carries some substantial disadvantages, what makes it necessary to nd a dierent way. Here, we explore some renowned results from real algebraic geometry, which has Stengle's positivstellensatz as one of its central pieces, and SOS optimization. Once the proper link is made, the original problem can be reduced to a nonlinear semidenite programming one, which has peculiarities that favours the application of an inexact restoration paradigm. We point out the algebraic view and the no use of discretizations as great advantages of this approach, besides the notable versatility and easy generalization in terms of dimension and involved objects.
 
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mitodiss.pdf (22.78 Mbytes)
Publishing Date
2018-11-23
 
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