Distributed Computing for Huge-Scale Aggregative Convex Programming

Concerning huge-scale aggregative convex programming of a linear objective subject to the affine constraints of equality and inequality and the quadratic constraints of inequality, convex and aggregatively computable, an algorithm is developed for its distributed computing.

The consensus with single common variable is used to partition the constraints into multi-consensus blocks, and the subblocks of each consensus block are employed to partition the primal variables into multiple sets of disjoint subvectors.

The global consensus constraints of equality and the original constraints are converted into the extended constraints of equality via slack variables to help initialize the algorithm.

The augmented Lagrangian, the proximal point method with double proximal terms or single, the block-coordinate Gauss-Seidel method, and ADMM are used to update the primal and slack variable sequences; descent models with built-in bounds are used to update the dual, motivated by the mathematical structures of the first-order characteristics of the update rules for the primal and slack.

The feasibility conditions for the algorithm to produce optimal solutions are described and their realizations through initial and parameter values are outlined.

Under the feasibility conditions supposed, convergence of the algorithm to optimal solutions is argued and the rate of convergence, $O(1/k^{1/2})$ is estimated roughly.

Issues requiring further explorations are listed.