Abstract.
This paper demonstrates that for generalized methods of multipliers for convex programming based on Bregman distance kernels – including the classical quadratic method of multipliers – the minimization of the augmented Lagrangian can be truncated using a simple, generally implementable stopping criterion based only on the norms of the primal iterate and the gradient (or a subgradient) of the augmented Lagrangian at that iterate. Previous results in this and related areas have required conditions that are much harder to verify, such as ε-optimality with respect to the augmented Lagrangian, or strong conditions on the convex program to be solved. Here, only existence of a KKT pair is required, and the convergence properties of the exact form of the method are preserved. The key new element in the analysis is the use of a full conjugate duality framework, as opposed to mainly examining the action of the method on the standard dual function of the convex program. An existence result for the iterates, stronger than those possible for the exact form of the algorithm, is also included.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: February 6, 2001 / Accepted: January 25, 2003 Published online: March 21, 2003
Mathematics Subject Classification (1991): 90C25, 90C46, 47H05
Rights and permissions
About this article
Cite this article
Eckstein, J. A practical general approximation criterion for methods of multipliers based on Bregman distances. Math. Program., Ser. A 96, 61–86 (2003). https://doi.org/10.1007/s10107-003-0374-x
Issue Date:
DOI: https://doi.org/10.1007/s10107-003-0374-x