A two-phase strategy for control constrained elliptic optimal control problems

Elliptic optimal control problems with pointwise box constraints on the control (EOCP) are considered. To solve EOCP, the primal-dual active set (PDAS) method, which is a special semismooth Newton (SSN) method, used to be a priority in consideration of their locally superlinear convergence. However, in general solving the Newton equations is expensive, especially when the discretization is in a fine level. Motivated by the success of applying alternating direction method of multipliers (ADMM) for solving large scale convex minimization problem in finite dimension, it is reasonable to extend the ADMM to solve EOCP. To numerically solve EOCP, the finite element (FE) method is used for discretization. Then, a two-phase strategy is presented to solve discretized problems. In Phase-I, an inexact heterogeneous ADMM (ihADMM) is proposed with the aim of solving discretized problems to moderate accuracy or using it to generate a reasonably good initial point to warm-start Phase-II. Different from the classical ADMM, our ihADMM adopts two different weighted inner product to define the augmented Lagrangian function in two subproblems, respectively. Benefiting from such different weighted techniques, two subproblems of ihADMM can be efficiently implemented. Furthermore, theoretical results on the global convergence as well as the iteration complexity results $o(1/k)$ for ihADMM are given. In Phase-II, in order to obtain more accurate solution, the primal-dual active set (PDAS) method is used as a postprocessor of the ihADMM. Numerical results show that the ihADMM and the two-phase strategy are highly efficient.