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In this paper we propose and investigate a new class of dual updates within the augmented Lagrangian framework, where the key feature is to reverse the update direction in the traditional dual ascent. When the dual variable is further scaled by a fractional number, we name the resulting scheme scaled dual descent (SDD), and otherwise, unscaled dual descent (UDD). The novel concept of dual descent sheds new light on the classic augmented Lagrangian framework and brings in more possibilities for algorithmic development. For nonlinear equality-constrained multi-block problems, we propose a new algorithm named SDD-ADMM, which combines SDD with the alternating direction method of multipliers (ADMM). We show that SDD-ADMM finds an ǫ-stationary solution in O(ǫ −4 ) iterations, which can be further improved to O(ǫ −3 ) under an additional technical while verifiable assumption. We also propose UDD-ALM, which applies UDD within the augmented Lagrangian method (ALM), for weakly convex minimization over affine constraints. We show that when a certain regularity condition holds at the primal limit point, UDD-ALM asymptotically converges to a stationary point and finds an ǫ-stationary solution in O(ǫ −2 ) iterations. Both SDD-ADMM and UDD-ALM are single-looped, and our iteration complexities are measured by first-order oracles. SDD-ADMM achieves better iteration complexities for a more challenging setting compared to the ADMM literature, while UDD-ALM complements the best-known result of existing ALM-base algorithms. Furthermore, we demonstrate the potential of UDD-ALM to handle nonlinear constraints by assuming a novel descent solution oracle for the proximal augmented Lagrangian relaxation.

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