  We contribute improvements to a Lagrangian dual solution approach applied to
large-scale optimization problems whose objective functions are convex,
continuously differentiable and possibly nonlinear, while the non-relaxed
constraint set is compact but not necessarily convex. Such problems arise, for
example, in the split-variable deterministic reformulation of stochastic
mixed-integer optimization problems. The dual solution approach needs to
address the nonconvexity of the non-relaxed constraint set while being
efficiently implementable in parallel. We adapt the augmented Lagrangian method
framework to address the presence of nonconvexity in the non-relaxed constraint
set and the need for efficient parallelization. The development of our approach
is most naturally compared with the development of proximal bundle methods and
especially with their use of serious step conditions. However, deviations from
these developments allow for an improvement in efficiency with which
parallelization can be utilized. Pivotal in our modification to the augmented
Lagrangian method is the use of an integration of approaches based on the
simplicial decomposition method (SDM) and the nonlinear block Gauss-Seidel (GS)
method. An adaptation of a serious step condition associated with proximal
bundle methods allows for the approximation tolerance to be automatically
adjusted. Under mild conditions optimal dual convergence is proven, and we
report computational results on test instances from the stochastic optimization
literature. We demonstrate improvement in parallel speedup over a baseline
parallel approach.
