and Gabay et al. in [@Gabay1976A; @Glowinski1975] and has been broadly used in many areas. Motivated by the success of ADMM in solving finite dimensional large scale optimization problem [@Chen2017; @Fazel2013; @Li2015; @Li2016], ADMM-type method has been used to solve PDE-constrained optimization problems in function space in [@Chen2018; @li2018efficient; @Song2018A; @zhang2017alternating]. While these works all focus on *First discretize, then optimize* method, and to the best of our knowledge, very little work has been done to apply *First optimize, then discretize* to solve PDE-constrained optimization problems by ADMM-type method. We apply ADMM as the outer optimization algorithm of the *‘optimize-discretize-optimize’* strategy, propose a new algorithm to solve PDE-constrained optimization problems efficiently and give the convergence analysis.

First, we briefly give the iterative format of the classical ADMM for the 2-block convex optimization problem with linear constraints: $$\begin{aligned} 
\min \limits_{(x,y)\in X\times Y}^{}&f(x)+g(y) \\
{\rm s.t.}\ \ \ \ &Ax+By=c,
\end{aligned}$$ where $X$,$Y$,$\Lambda$ are finite dimensional real Euclidean spaces, $f(x):X\rightarrow (-\infty, +\infty]$, $g(y):Y\rightarrow (-\infty, +\infty]$ are convex functions (maybe nonsmooth), $A: X\rightarrow \Lambda$, $B: Y\rightarrow \Lambda$ are linear mappings, $c \in \Lambda$ is given. The augmented Lagrangian function is given by the following form: $$\mathcal{L}_{\rho}(x,y,\lambda;\rho)=f(x)+g(y)+(\lambda,Ax+By-c)+\frac{\rho}{2}\|Ax+By-c\|^2,$$ where $\lambda \in \Lambda$ denotes the Lagrange 