Auxiliary Problem Principle of augmented Lagrangian with Varying Core Functions for Large-Scale Structured Convex Problems

Abstract: The auxiliary problem principle of augmented Lagrangian (APP-AL), proposed by Cohen and Zhu (1984), aims to find the solution of a constrained optimization problem through a sequence of auxiliary problems involving augmented Lagrangian. The merits of this approach are two folds. First, the core function is usually separable, which makes the subproblems at each step decomposable and particularly attractive for parallel computing. Second, the choice of the core function is quite flexible. Consequently, by carefully specifying this function, APP-AL may reduce to some standard optimization algorithms. In this paper, we pursue enhancing such flexibility by allowing the core function to be non-identical at each step of the algorithm, and name it varying auxiliary problem principle (VAPP-AL). Depending on the problem structure, the varying core functions in VAPP-AL can be adapted to design new flexible and suitable algorithm for parallel and distributed computing. The convergence and O(1/t) convergence rate of VAPP-AL for convex problem with coupling objective and constraints is proved. Moreover, if this function is specialized to be quadratic, an o(1/t) convergence rate can be established. Interestingly, the new VAPP framework can cover several variants of Jacobian type augmented Lagrangian decomposition methods as special cases. Furthermore, our technique works for the convex problem with nonseparable objective and multi-blocks coupled linear constraints, which usually can not be handled by ADMM.