Parallel Variable Distribution for Constrained Optimization

Abstract

In the parallel variable distribution framework for solving optimization problems (PVD), the variables are distributed among parallel processors with each processor having the primary responsibility for updating its block of variables while allowing the remaining “secondary” variables to change in a restricted fashion along some easily computable directions. For constrained nonlinear programs convergence theory for PVD algorithms was previously available only for the case of convex feasible set. Additionally, one either had to assume that constraints are block-separable, or to use exact projected gradient directions for the change of secondary variables. In this paper, we propose two new variants of PVD for the constrained case. Without assuming convexity of constraints, but assuming block-separable structure, we show that PVD subproblems can be solved inexactly by solving their quadratic programming approximations. This extends PVD to nonconvex (separable) feasible sets, and provides a constructive practical way of solving the parallel subproblems. For inseparable constraints, but assuming convexity, we develop a PVD method based on suitable approximate projected gradient directions. The approximation criterion is based on a certain error bound result, and it is readily implementable. Using such approximate directions may be especially useful when the projection operation is computationally expensive.