Methods for vector optimization: trust region and proximal on riemannian manifolds and Newton with variable order
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2017-08-28
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Universidade Federal de Goiás
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In this work, we will analyze three types of method to solve vector optimization problems
in different types of context. First, we will present the trust region method for multiobjective
optimization in the Riemannian context, which retrieves the classical trust region method for
minimizing scalar functions. Under mild assumptions, we will show that each accumulation
point of the generated sequences by the method, if any, is Pareto critical. Next, the proximal
point method for vector optimization and its inexact version will be extended from Euclidean
space to the Riemannian context. Under suitable assumptions on the objective function,
the well-definedness of the methods will be established. Besides, the convergence of any
generated sequence, to a weak efficient point, will be obtained. The last method to be
investigated is the Newton method to solve vector optimization problem with respect to
variable ordering structure. Variable ordering structures are set-valued map with cone values
that to each element associates an ordering. In this analyze we will prove the convergence
of the sequence generated by the algorithm of Newton method and, moreover, we also will
obtain the rate of convergence under variable ordering structures satisfying mild hypothesis.
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PEREIRA, Y. R. L. Methods for vector optimization: trust region and proximal on riemannian manifolds and Newton with variable order. 2017. 62 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2017.