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duality

# duality - EE236A(Fall 2007-08 Lecture 9 Duality(part 1 •...

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Unformatted text preview: EE236A (Fall 2007-08) 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 9–1 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) 9–2 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) 9–3 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) 9–4 Piecewise-linear minimization minimize max i =1 ,...,m ( a T i x − b i ) lower bounds for optimal value p ⋆ ?...
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duality - EE236A(Fall 2007-08 Lecture 9 Duality(part 1 •...

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