# Consider a ‘‘nested decomposition’’ as applied to the problem

Consider a ‘‘nested decomposition’’ as applied to the problem

Let (1) be the constraints of the (first) restricted master problem. If πi (i = 1, 2, . . . , k) are shadow prices for the constraints (1) in the weighting problem, then

constitutes subproblem 1 (the proposal-generating problem). Suppose, though, that the constraints (3 0 ) complicate this problem and make it difficult to solve. Therefore, to solve the subproblem we further apply decomposition on subproblem 1. Constraints (2 0 ) will be the constraints of the ‘‘second’’ restricted master. Given any shadow prices αi(i = k + 1, k + 2, . . . , `) for constraints (2 0 ) in the weighting problem, the subproblem 2 will be:

a) Consider the following decomposition approach: Given shadow prices πi , solve subproblem (1) to completion by applying decomposition with subproblem (2). Use the solution to this problem to generate a new weighting variable to the first restricted master problem, or show that the original problem (P) [containing all constraints (1), (2), (3)] has been solved. Specify details of this approach

. b) Show finite convergence and bounds on the objective function to (P).

c) Now consider another approach: Subproblem 1 need not be solved to completion, but merely until a solution x j (j = 1, 2, . . . , n) is found, so that

where γ is the shadow price for the weighting constraint to the first restricted master. Indicate how to identify such a solution x j (j = 1, 2, . . . , n) while solving the second restricted master problem; justify this approach.

d) Discuss convergence and objective bounds for the algorithm proposed in part (c).