  Recently, a number of learning-based optimization methods that combine
data-driven architectures with the classical optimization algorithms have been
proposed and explored, showing superior empirical performance in solving
various ill-posed inverse problems, but there is still a scarcity of rigorous
analysis about the convergence behaviors of learning-based optimization. In
particular, most existing analyses are specific to unconstrained problems but
cannot apply to the more general cases where some variables of interest are
subject to certain constraints. In this paper, we propose Differentiable
Linearized ADMM (D-LADMM) for solving the problems with linear constraints.
Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which is
obtained by firstly introducing some learnable weights in the classical
Linearized ADMM algorithm and then generalizing the proximal operator to some
learnable activation function. Notably, we rigorously prove that there exist a
set of learnable parameters for D-LADMM to generate globally converged
solutions, and we show that those desired parameters can be attained by
training D-LADMM in a proper way. To the best of our knowledge, we are the
first to provide the convergence analysis for the learning-based optimization
method on constrained problems.
