Tesis Doctorado
Contribuciónes a la programación cónica de segundo orden y a la programación matricial usando métodos de métrica variable
Autor
Alvarez-Daziano, Felipe
Ramírez-Cabrera, Héctor
Universidad de Chile
Institución
Resumen
This dissertation is organized as follows:
Chapter 1: In this introductory chapter, we expose briefiy the results that we have obtained.
In each section, we present the main problem that we have considered, give the necessary
definitions that help to understand the subject, recall the known results in the literature
about the problem, and finally, describe our contribution to the subject. For a more complete
description, we refer the reader to corresponding chapters.
Chapter 2: This chapter deals with a variable metric interior proximal type algorithm for
solving convex second-order cone programs, which · is induced by a class of positive definite
matrices. This algorithm uses an appropriate choice of a regularizat10n parameter in order
to ensure the well-definedness of the proximal algorithm and to force the iterates to belong
to the interior of the fe asible seto We provide conditions that guarantee convergence of t he
proximal sequence. To illustrate how our algorithm works in practice, numerical tests applied
to structural optimization and support vector machines, are presented. The results
in this chapter were obtained in collaboration with Dr. F. Alvarez and Dr. H. Ramírez,
and are contained in the research paper Interior Proximal Algorithm with Variable Metric
for Second-Order Gone Programming: Applications to Structural Optimization and Support
Vector Machines [11], submitted for publication in Optimization Methods and Software.
Chapter 3: Here we study of the dual and primal-dual central paths associated with
penalty jbarrier functions for solving semidefinite programming problems. These penalty jbarrier
functions are computed as spectrally defined functions generated by a real penalty j barrier
functions. We establish the convergence of this central path. In addition, we study the
existence of Cauchy trajectories in semidefinite programming and we also prove the value
convergence of a given objective function f. Sorne examples of Cauchy trajectories are presented.
The results in this chapter were obtained in collaboration with Dr. F. Alvarez and Dr. H. RamÍrez.