First-Order Algorithms for Convex Optimization with Nonseparable Objective and Coupled Constraints

In this paper, we consider a block-structured convex optimization model, where in the objective the block variables are nonseparable and they are further linearly coupled in the constraint. For the 2-block case, we propose a number of first-order algorithms to solve this model. First, the alternating direction method of multipliers (ADMM) is extended, assuming that it is easy to optimize the augmented Lagrangian function with one block of variables at each time while fixing the other block. We prove that O(1/t) iteration complexity bound holds under suitable conditions, where t is the number of iterations. If the subroutines of the ADMM cannot be implemented, then we propose new alternative algorithms to be called alternating proximal gradient method of multipliers, alternating gradient projection method of multipliers, and the hybrids thereof. Under suitable conditions, the O(1/t) iteration complexity bound is shown to hold for all the newly proposed algorithms. Finally, we extend the analysis for the ADMM to the general multi-block case.