Inexact variants of the alternating direction method of multipliers and their iteration-complexity analyses

• Autor(a) principal: Adona, Vando Antônio
• Data de Publicação: 2019
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFG
• dARK ID: ark:/38995/0013000009tj0
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/9522
• topic: Método dos multiplicadores das direções alternadas; Programa convexo; Método extragradiente híbrido; Critério de erro relativo; Iteração complexidade pontual; Iteração complexidade ergódica; Alternating direction method of multipliers; Convex program; Hybrid extragradient method; Relative error criterion; Pointwise iteration-complexity; Ergodic iteration-complexity; CIENCIAS EXATAS E DA TERRA::MATEMATICA

This thesis proposes and analyzes some variants of the alternating direction method of multipliers (ADMM) for solving separable linearly constrained convex optimization problems. This thesis is divided into three parts. First, we establish the iteration-complexity of a proximal generalized ADMM. This ADMM variant, proposed by Bertsekas and Eckstein, introduces a relaxation parameter into the second ADMM subproblem in order to improve its computational performance. We show that, for a given tolerance ρ>0, the proximal generalized ADMM with α in (0, 2) provides, in at most O(1/ρ^2) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most O(1/ρ) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the proximal generalized ADMM with α in (0, 2]. Second, we propose and analyze an inexact variant of the aforementioned proximal generalized ADMM. In this variant, the rst subproblem is approximately solved using a relative error condition whereas the second one is assumed to be easy to solve. It is important to mention that in many ADMM applications one of the subproblems has a closed-form solution; for instance, l_1-regularized convex composite optimization problems. We show that the proposed method possesses iteration-complexity bounds similar to its exact version. Third, we develop an inexact proximal ADMM whose rst subproblem is inexactly solved using an approximate relative error criterion similar to the aforementioned inexact proximal generalized ADMM. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. Our approach consists of interpreting these ADMM variants as an instance of a hybrid proximal extragradient framework with some special properties. Finally, in order to show the applicability and advantage of the inexact ADMM variants proposed here, we present some numerical experiments performed on a setting of problems derived from real-life applications.