Numerical Solution for Sparse PDE Constrained Optimization

In this chapter, elliptic PDE-constrained optimal control problems with L1-control cost (L1-EOCP) are considered. Motivated by the success of the first-order methods, we give an overview on two efficient first-order methods to solve L1-EOCP: inexact heterogeneous alternating direction method of multipliers (ihADMM) and an inexact symmetric Gauss-Seidel (sGS)-based 2-block majorized accelerated block coordinate descent (ABCD) method (sGS-imABCD). Different from the classical ADMM, the ihADMM adopts two different weighted inner products 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. A common approach to solve the L1-EOCP is directly solving the primal problem. Based on the dual problem of L1-EOCP, which can be reformulated as a multi-block unconstrained convex composite minimization problem, an efficient inexact ABCD method is introduced for solving L1-EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact sGS technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block.