Convex Optimization by S. Boyd, L. Vandenberghe

By S. Boyd, L. Vandenberghe

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The theory is based on axioms and apportionment methods presented in this chapter. The axiomatic method of apportionment theory relies on some socially desirable characteristics that one requires an apportionment to posses. These characteristics include, for instance house and population monotonicity but also many others. They have been shown crucial for the solutions of the apportionment problem. The famous Impossibility Theorem of Balinski and Young puts a clear limitation on which characteristics do not contradict one another by showing that having solutions that satisfy quota and that are at the same time population monotone is generally impossible.

8) where by definition Y0i = 0 and Ydii +1 − 1 = D. 3 | 0 − kri | 2 | 1 − kri | 1 | 2 − kri | | 3 − kri | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k Fig. 3 The Transformation to the Assignment Problem 37 Now, suppose that πi is not given but rather we would like to find out a πi = {Y1i , . 8) in isolation from all other j = i. 9) subject to i ≥ Y ji + 1, j = 1, . , di − 1, Y j+1 1 ≤ Y ji ≤ D, j = 1, . , di , Y ji are non-negative integers. 10) It turns out that obtaining an optimal solution for (P2i) is quite straightforward.

28) Proof. The proof is by induction on di α (πi ) = ∑ Y ji − Z ij . j=1 For πi with α = 0, we have Y ji = Z ij , for j = 1, .

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