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08-decomposition_notes

# 08-decomposition_notes - Notes on Decomposition Methods...

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Unformatted text preview: Notes on Decomposition Methods Stephen Boyd, Lin Xiao, Almir Mutapcic, and Jacob Mattingley Notes for EE364B, Stanford University, Winter 2006-07 April 13, 2008 Contents 1 Primal decomposition 3 1.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Dual decomposition 7 2.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Decomposition with constraints 11 3.1 Primal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Dual decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Coupling constraints and coupling variables . . . . . . . . . . . . . . . . . . 15 4 More general decomposition structures 17 4.1 General examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Framework for decomposition structures . . . . . . . . . . . . . . . . . . . . 19 4.3 Primal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Dual decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Rate control 24 5.1 Dual decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6 Single commodity network flow 28 6.1 Dual decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.2 Analogy with electrical networks . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1 Decomposition is a general approach to solving a problem by breaking it up into smaller ones and solving each of the smaller ones separately, either in parallel or sequentially. (When it is done sequentially, the advantage comes from the fact that problem complexity grows more than linearly.) Problems for which decomposition works in one step are called (block) separable , or trivially parallelizable . As a general example of such a problem, suppose the variable x can be partitioned into subvectors x 1 , . . ., x k , the objective is a sum of functions of x i , and each constraint involves only variables from one of the subvectors x i . Then evidently we can solve each problem involving x i separately (and in parallel), and then re-assemble the solution x . Of course this is a trivial, and not too interesting, case. A more interesting situation occurs when there is some coupling or interaction between the subvectors, so the problems cannot be solved independently. For these cases there are techniques that solve the overall problem by iteratively solving a sequence of smaller problems. There are many ways to do this; in this note we consider some simple examples that illustrate the ideas....
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08-decomposition_notes - Notes on Decomposition Methods...

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