Globally Convergent Distributed Sequential Quadratic Programming with Overlapping Decomposition and Exact Augmented Lagrangian Merit Function

In this paper, we address the problem of solving large-scale graph-structured nonlinear programs (gsNLPs) in a scalable manner. GsNLPs are problems in which the objective and constraint functions are associated with a graph node, and they depend only on the variables of adjacent nodes. This graph-structured formulation encompasses various specific instances, such as dynamic optimization, PDE-constrained optimization, multi-stage stochastic optimization, and optimization over networks. We propose a globally convergent overlapping graph decomposition method for solving large-scale gsNLPs under the standard regularity assumptions and mild conditions on the graph topology. At each iteration step, we use an overlapping graph decomposition to compute an approximate Newton step direction using parallel computations. We then select a suitable step size and update the primal-dual iterates by performing backtracking line search with the exact augmented Lagrangian merit function. By exploiting the exponential decay of sensitivity of gsNLPs, we show that the approximate Newton direction is a descent direction of the augmented Lagrangian merit function, which leads to global convergence and fast local convergence. In particular, global convergence is achieved for sufficiently large overlaps, and the local linear convergence rate improves exponentially in terms of the overlap size. This result matches existing results for dynamic programs. We validate our theory with an elliptic PDE-constrained problem.