Boundedness of dual iterates

Establish boundedness of the dual iterates generated by the Bregman proximal linearized ADMM for nonconvex composite-structured fractional optimization problems with nonlinear coupling constraints, without imposing boundedness of the full generated sequence as an assumption.

Background

The convergence analysis of the Bregman proximal linearized ADMM assumes that the complete sequence of primal and dual iterates is bounded. The paper notes that coercivity or level-boundedness of the augmented Lagrangian can ensure boundedness of the primal iterates, but does not provide a corresponding argument for the multiplier sequence.

Consequently, the boundedness of the dual iterates is left unresolved for the considered fractional optimization model with nonlinear coupling constraints. Resolving this issue would weaken the standing bounded-iterate assumption and broaden the applicability of the convergence results.

References

Dual iterate boundedness remains open and is not addressed in our work.

— A Bregman proximal linearized ADMM for fractional programming with nonlinear coupling constraints  (2610.11961 - Liu et al., 8 Oct 2026) in Remark following Lemma 3.3, Section 3.2, “Preliminary convergence results”