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Unformatted text preview: EE236A (Fall 200708) Lecture 9 Duality (part 1) the dual of an LP in inequality form weak duality examples optimality conditions and complementary slackness Farkas lemma and theorems of alternatives proof of strong duality 91 The dual of an LP in inequality form LP in inequality form: minimize c T x subject to Ax b n variables, m inequality constraints, optimal value p called primal problem (in context of duality) the dual LP : maximize b T z subject to A T z + c = 0 z an LP in standard form with m variables, n equality constraints optimal value denoted d main property : p = d (if primal or dual is feasible) Duality (part 1) 92 Weak duality if x is primal feasible and z is dual feasible, then c T x b T z proof : Ax b , A T z + c = 0 , z imply c T x = z T Ax z T b c T x + b T z is called the duality gap associated with x and z weak duality : minimize l.h.s. over x , maximize r.h.s. over z : p d always true (even when p = + and/or d = ) Duality (part 1) 93 Example minimize 4 x 1 5 x 2 subject to 1 2 1 1 1 2 bracketleftbigg x 1 x 2 bracketrightbigg 3 3 x = (1 , 1) with objective value 9 dual problem maximize 3 z 2 3 z 4 subject to bracketleftbigg 1 2 1 1 1 2 bracketrightbigg z 1 z 2 z 3 z 4 + bracketleftbigg 4 5 bracketrightbigg = 0 z 1 , z 2 , z 3 , z 4 z = (0 , 1 , , 2) is dual feasible with objective value 9 conclusion (by weak duality): x is optimal Duality (part 1) 94 Piecewiselinear minimization minimize max i =1 ,...,m ( a T i x b i ) lower bounds for optimal value p ?...
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 Spring '07
 Dr.Vandenber

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