Dantzig-Wolfe Decomposition with GAMS - Amsterdam Optimization ...

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4 ERWIN KALVELAGENSub 1DualsSub 2MasterNew columns.Sub KFigure 1. Communication between restricted master and sub-problemsThe attractiveness of a variable λ k,j can be measured by its reduced cost 1 . Ifwe denote the dual variables for constraint B 0 x 0 + Bλ ′ = b 0 by π 1 and those forthe convexity constraints ∑ j δ k,jλ ′ k,j= 1 by π(k) 2 , then reduced cost for the masterproblem look like:((14) σ k,j = (c T k x (j)k) − Bk x (j) )πT k = (c T k − π1 T B k )x (j)kδ − π(k) 2 δ k,jk,jAssuming the sub-problem to be bounded, the most attractive bfs (basic feasiblesolution) x k to enter the master problem is found by maximizing the reduced costgiving the following LP:(15)min σ k = (c T k − π1 T B k )x k − π (k)x2kB k x x = b kx k ≥ 0The operation to find these reduced costs is often called Pricing. If σk ∗ < 0 we canintroduce the a new column λ k,j to the master, with a cost coefficient of c T k x∗ k .A basic Dantzig-Wolfe decomposition algorithm can now be formulated:Dantzig-Wolfe decomposition algorithm.{initialization}Choose initial subsets of variables.while true do1 The reduced cost of a variable xj is(13) σ j = c j − π T A jwhere A j is the column of A corresponding to variable x j , and π are the duals.
DANTZIG-WOLFE DECOMPOSITION WITH GAMS 5{Master problem}Solve the restricted master problem.π 1 := duals of coupling constraintsπ (k)2 := duals of the k th convexity constraint{Sub problems}for k=1,. . . ,K doPlug π 1 and π (k)2 into sub-problem kSolve sub-problem kif σk ∗ < 0 thenAdd proposal x ∗ kto the restricted masterend ifend forif No proposals generated thenStop: optimalend ifend while(16)5. InitializationWe did not pay attention to the initialization of the decomposition.The first thing we can do is solve each sub-problem:min c T k x kA k x k = b kx k ≥ 0If any of the subproblems is infeasible, the original problem is infeasible. Otherwise,we can use the optimal values x ∗ k(or the unbounded rays) to generate an initial setof proposals.6. Phase I/II algorithmThe initial proposals may violate the coupling constraints. We can formulatea Phase I problem by introducing artificial variables and minimizing those. Theuse of artificial variables is explained in any textbook on Linear Programming (e.g.[1, 14]). It is noted that the reduced cost for a Phase I problem are slightly differentfrom the Phase II problem.As an example consider that the coupling constraints are∑(17)x j ≤ bWe can add an artificial variable x a ≥ 0 as follows:∑(18)x j − x a ≤ bThe phase I objective will be:(19) min x ajjThe reduced cost of a variable x j is now as in equation (14) but with c T k = 0.It is noted that it is important to remove artificials once a phase II starts. Wedo this in the example code by fixing the artificial variables to zero.
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