  We consider a two-stage stochastic optimization problem, in which a long-term
optimization variable is coupled with a set of short-term optimization
variables in both objective and constraint functions. Despite that two-stage
stochastic optimization plays a critical role in various engineering and
scientific applications, there still lack efficient algorithms, especially when
the long-term and short-term variables are coupled in the constraints. To
overcome the challenge caused by tightly coupled stochastic constraints, we
first establish a two-stage primal-dual decomposition (PDD) method to decompose
the two-stage problem into a long-term problem and a family of short-term
subproblems. Then we propose a PDD-based stochastic successive convex
approximation (PDD-SSCA) algorithmic framework to find KKT solutions for
two-stage stochastic optimization problems. At each iteration, PDD-SSCA first
runs a short-term sub-algorithm to find stationary points of the short-term
subproblems associated with a mini-batch of the state samples. Then it
constructs a convex surrogate for the long-term problem based on the deep
unrolling of the short-term sub-algorithm and the back propagation method.
Finally, the optimal solution of the convex surrogate problem is solved to
generate the next iterate. We establish the almost sure convergence of PDD-SSCA
and customize the algorithmic framework to solve two important application
problems. Simulations show that PDD-SSCA can achieve superior performance over
existing solutions.
