Tesis de Postgrado Acceso Abierto
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Thesis Primal-dual splitting algorithms for constrained monotone inclusions(2020) Sergio Eduardo, López Rivera; Deride Silva, Julio; Briceño Arias, LuisIn this thesis we propose an efficient splitting algorithm for solving constrained primal-dual monotone inclusions with a normal cone to a closed vector subspace. Our algorithm incorporates an additional projection onto a set of constraints, which represents a priori information on the solutions. This work is divided in two parts. In the first part, we study the case when the vector subspace is the whole space, in which we provide weak convergence of our method, as well as accelerated convergence and linear convergence under corresponding additional hypotheses on the operators and step sizes of the algorithm. In the second part, we consider the general case and we demonstrate weak convergence of our method by characterizing the solutions to the inclusión using the partial inverse operator. The efficiency of our method is illustrated in the context of convex optimization with affine linear constraints and vector subspace constraints. The use of the a priori information allows the feasibility of primal iterations in a subset of constraints and the use of the partial inverse operator allows to exploit the vector subspace structure. These two features of our method improve the efficiency with respect to several methods in the literature. Finally, we also apply our method to solve the traffic assignment problem with arc-capacity expansion on a network with minimal cost under uncertainty, in which we show the advantage of using the vector subspace structure of the problem.
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