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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 Piecewise-linear minimization minimize max i =1 ,...,m ( a T i x b i ) lower bounds for optimal value p ?...
View Full Document

Page1 / 9

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online