Primal Decomposition of Mathematical Programs by Resource Allocation: I—Basic Theory and a Direction-Finding Procedure

This paper presents a method for primal decomposition of large convex separable programs into a sequence of smaller subproblems. The main advantage of primal decomposition over Lagrange-multiplier or dual-decomposition methods is that a primal feasible solution is maintained during the course of the iterations. Feasibility is maintained by recasting the original convex separable program into a context of resource allocation. Then a direction-finding procedure for a method of feasible directions is developed for the derived resource-allocation problem. The direction-finding procedure utilizes the directional derivative to give a piecewise-linear approximation to the primal resource-allocation function. This approximation is more efficient than the usual linear gradient approximation used in methods of feasible direction, but it is still made with a linear program. The efficiency of the piecewise-linear approximation and the operation of the method of feasible directions are illustrated by a simple numerical example.