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in this algorithm, both subproblems are in equivalent positions and play the role of both
the master problem and the subproblem of the traditional method At each iteration, the stage one
subproblem gives the upper bound of the onginai pmblem by restncting the dual variables to
convex combinations of known dual solutions and the stage two subproblem gives a lower bound
by restricting primal vviables to convex combinations of known primal solutions. The algorithm
itemtes until the upper bound and Iower bound reach an equilibnum point.
Their tests for eleven problerns with the two stage, block-uiangular stmcnires showed bat
the new method is usually faster and more efficient than the iraditional rnethods.
Park [19%] extended this idea to convex. nonlinear protgmming rnodeis. He proved and
tested that the new algorithm for the two stage case converged to an optimal solution in a finite
number of i tentions.
Prirnal-Dual Nested Decornposition Method for Multi-Stage Models
Lm and Fuller [199Sb] also presented a nested primal-dual decomposition aigorithm for
the multi-stage LP problerns with block-trianplar matrix structure. In ihis aipnthrn. the originiil
multi-stage problem is divided into a sequence of a pair of subproblems for each stage. These
subproblems are coordinated by passing the proposals forward and cuts backwud; the previous
subprobIems pass the proposals to the following subproblerns fonvard in stage numben (which
designates time period in many models) time and the following subproblems provide cuts to the
previous subproblems backwud. This information fiow between subproblerns is shown in Figure
2.1. from h and Fuller [1995b]. The aigorithm can be perceived as the combination of the
primd and the dual nested decomposition algorithms. As in the two stage case, the fint
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