For a linear programming problem z*= min {cx : Ax m b, p [ x [ q}, consider the following question: Given d > 0, describe the set, S, of column vectors, with their corresponding costs, such that any one of these columns when introduced in the constraint matrix A guarantees a decrease (over that of z*) in the optimum solution of the new linear program by at least d . This and related questions arise while designing an iterative allocation scheme in which a single buyer wants to acquire a set of items by soliciting bids from multiple suppliers via a competitive bidding process.
In this note, we characterize the set S and show that the separation problem over S can be solved in polynomial time. This is then used to solve a related optimization problem in each iteration of the bidding process.
By: M. Dawande, R. Chandrasekaran, J. Kalagnanam
Published in: RC22559 in 2002
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