The Decomposition Principle and Algorithms for Linear Programming

The computational difficulties that continue to plague decomposition algorithms, namely, “long-tail” convergence and numerical instabilities, have served to dampen enthusiasm about their computational effectiveness. The use of interior points of subproblems in decomposition procedures may have a significant role to play in alleviating such computational difficulties. Indeed, Dantzig-Wolfe decomposition provides the arena within which simplex techniques for master problems and interior-point techniques for subproblems complement one another in a useful way. In combination they could lead to more effective decomposition algorithms than we have today. We formulate a particular algorithm along these lines and illustrate its convergence and numerical characteristics through numerical experiments. We make these experiments the basis for a discussion of the merits of using interior points in decomposition.