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The covering problem via convex algebraic geometry

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Author(s):
Leonardo Makoto Mito
Total Authors: 1
Document type: Master's Dissertation
Press: São Paulo.
Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
Defense date:
Examining board members:
Gabriel Haeser; Marcelo Dias Passos; Thadeu Alves Senne
Advisor: Gabriel Haeser
Abstract

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. (AU)

FAPESP's process: 16/16999-5 - The sphere covering problem via convex algebraic geometry
Grantee:Leonardo Makoto Mito
Support Opportunities: Scholarships in Brazil - Master