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Optimal proximal augmented Lagrangian method and its application to full Jacobian splitting for multi-block separable convex minimization problems
- Source :
- IMA Journal of Numerical Analysis. 40:1188-1216
- Publication Year :
- 2019
- Publisher :
- Oxford University Press (OUP), 2019.
-
Abstract
- The augmented Lagrangian method (ALM) is fundamental in solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper we show that it is not necessary to employ a positive-definite quadratic proximal term for the proximal ALM and the convergence can be still ensured if the positive definiteness is relaxed to indefiniteness by reducing the proximal parameter. An indefinite proximal version of the ALM is thus proposed for the generic setting of convex programming problems with linear constraints. We show that our relaxation is optimal in the sense that the proximal parameter cannot be further reduced. The consideration of indefinite proximal regularization is particularly meaningful for generating larger step sizes in solving ALM’s primal subproblems. When the model under discussion is separable in the sense that its objective function consists of finitely many additive function components without coupled variables, it is desired to decompose each ALM’s subproblem over the primal variable in Jacobian manner, replacing the original one by a sequence of easier and smaller decomposed subproblems, so that parallel computation can be applied. This full Jacobian splitting version of the ALM is known to be not necessarily convergent, and it has been studied in the literature that its convergence can be ensured if all the decomposed subproblems are further regularized by sufficiently large proximal terms. But how small the proximal parameter could be is still open. The other purpose of this paper is to show the smallest proximal parameter for the full Jacobian splitting version of ALM for solving multi-block separable convex minimization models.
- Subjects :
- 021103 operations research
Augmented Lagrangian method
Applied Mathematics
General Mathematics
0211 other engineering and technologies
010103 numerical & computational mathematics
02 engineering and technology
Computer Science::Numerical Analysis
01 natural sciences
Separable space
Computational Mathematics
symbols.namesake
Block (telecommunications)
Jacobian matrix and determinant
Convex optimization
symbols
Applied mathematics
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14643642 and 02724979
- Volume :
- 40
- Database :
- OpenAIRE
- Journal :
- IMA Journal of Numerical Analysis
- Accession number :
- edsair.doi...........567e9050ab29b2f3e2573de976c1ffcb