We revise the volume algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a ...
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We revise the volume algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null steps philosophy of bundle methods, VA produces green, yellow or red steps. In order to give convergence results, we introduce in VA a precise measure for the improvement needed to declare a green or serious step. This addition yields a revised formulation (RVA) that is halfway between VA and a specific bundle method, that we call BVA. We analyze the convergence properties of both RVA and BVA. Finally, we compare the performance of the modified algorithms versus VA on a set of Rectilinear Steiner problems of various sizes and increasing complexity, derived from real world VLSI design instances.
We deal with the linear programming relaxation of set partitioning problems arising in airline crew scheduling. Some of these linear programs have been extremely difficult to solve with the traditional algorithms, We ...
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We deal with the linear programming relaxation of set partitioning problems arising in airline crew scheduling. Some of these linear programs have been extremely difficult to solve with the traditional algorithms, We have used an extension of the subgradient algorithm, the volume algorithm, to produce primal solutions that might violate the constraints by at most 2%, and that are within 1% of the lower bound. This method is fast, requires minimal storage, and can be parallelized in a straightforward way. (C) 2002 Elsevier Science B.V. All rights reserved.
It is shown that the "hit-and-run" algorithm for sampling from a convex body K (introduced by R.L. Smith) mixes in time O*(n(2)R(2)/r(2)), where R and r are the radii of the inscribed and circumscribed balls...
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It is shown that the "hit-and-run" algorithm for sampling from a convex body K (introduced by R.L. Smith) mixes in time O*(n(2)R(2)/r(2)), where R and r are the radii of the inscribed and circumscribed balls of K. Thus after appropriate preprocessing, hit-and-run produces an approximately uniformly distributed sample point in time O*(n(3)), which matches the best known bound for other sampling algorithms. We show that the bound is best possible in terms of R, r and n.
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