Vector and matrix apportionment problems and separable convex integer optimization

N. Gaffke () and F. Pukelsheim ()

Mathematical Methods of Operations Research, 2008, vol. 67, issue 1, 133-159

Abstract: The problems of (bi-)proportional rounding of a nonnegative vector or matrix, resp., are written as particular separable convex integer minimization problems. Allowing any convex (separable) objective function we use the notions of vector and matrix apportionment problems. As a broader class of problems we consider separable convex integer minimization under linear equality restrictions Ax = b with any totally unimodular coefficient matrix A. By the total unimodularity Fenchel duality applies, despite the integer restrictions of the variables. The biproportional algorithm of Balinski and Demange (Math Program 45:193–210, 1989) is generalized and derives from the dual optimization problem. Also, a primal augmentation algorithm is stated. Finally, for the smaller class of matrix apportionment problems we discuss the alternating scaling algorithm, which is a discrete variant of the well-known Iterative Proportional Fitting procedure. Copyright Springer-Verlag 2008

Keywords: Proportional and biproportional rounding; Totally unimodular matrix; Elementary vector; Graver basis; Convex programming duality; Alternating maximization procedure (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s00186-007-0184-7

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