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Summsry: Let $7; c?enote the i~~f i m u n : of Lhe Integral cf z function w. r. t. rr!! probnbi!ity measures with given marginals. The determination of rn is of interest for a series of stochastic problems. In the present paper we prove a duality theorem for the determination nf 3. an(! give ~'xnmples fnr i t s rt.pplicet,ion, 7.Ve consirler especislly the problem of extrei-fia; variance of of ran&jm variables and prove a, iheorerL f L r iha esisLenc;: i;f rttndorn variables with given marginal distributions, such t h a t their sum has variance zero.

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Algorithms for the proportional rounding of a nonnegative vector, and for the biproportional rounding of a nonnegative matrix are discussed. Here we view vector and matrix rounding as special instances of a generic optimization problem that employs an additive version of the objective function of Gaffke and Pukelsheim (2007). The generic problem turns out to be a separable convex integer optimization problem, in which the linear equality constraints are given by a totally unimodular coefficient matrix. So, despite the integer restrictions of the variables, Fenchel duality applies. Our chief goal is to study the implied algorithmic consequences. We establish a general algorithm based on the primal optimization problem. Furthermore we show that the biproportional algorithm of Balinski and Demange (1989), when suitably generalized, derives from the dual optimization problem. Finally we comment on the shortcomings of the alternating scaling algorithm, a discrete variant of the well-known Iterative Proportional Fitting procedure.Short title: Apportionment and separable integer optimization.

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