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Unformatted text preview: Math 171A: Linear Programming Lecture 3 Geometry of the Feasible Region Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, January 7th, 2011 UCSD Center for Computational Mathematics Slide 1/47, Friday, January 7th, 2011 Recap: a linear inequality constraint a T x < b a T x > b a T x = b x 2 x 1 Recap: distance of a hyperplane to the origin x 2 x 1 a a T x = b k d k =  b  k a k d Recap: Properties of hyperplanes The closest distance of the hyperplane a T x = b to the origin is  b  k a k , where k a k = n X i =1 a 2 i 1 2 UCSD Center for Computational Mathematics Slide 4/47, Friday, January 7th, 2011 Distance of a point to a hyperplane x 2 x 1 ¯ x a Distance of a point to a hyperplane Result Given ¯ x ∈ R n and a hyperplane a T x = b , the quantity  a T ¯ x b  k a k measures the perpendicular distance of ¯ x to a T x = b . Proof: Main idea: Move the origin to ¯ x (Change coordinates from “ x ” to “ x ” using x = x ¯ x ) Find the distance of x = 0 to the hyperplane UCSD Center for Computational Mathematics Slide 6/47, Friday, January 7th, 2011 x 2 x 1 ¯ x a x 2 x 1 a x 2 x 1 ¯ x Change coordinates from “ x ” to “ x ” using x = x ¯ x . The hyperplane in the new coordinates is b = a T x = a T ( x + ¯ x ) = a T x + a T ¯ x ⇒ a T x = b a T ¯ x ⇒ a T x = b for b = b a T ¯ x Distance to the origin ( x = 0) is  b  k a k =  b a T ¯ x  k a k . UCSD Center for Computational Mathematics Slide 9/47, Friday, January 7th, 2011 Definition Two constraints are equivalent if they have the same set of feasible points. Examples: a T x ≥ b is equivalent to ( γ a ) T x ≥ γ b for γ > a T x ≤ b is equivalent to ( γ a ) T x ≥ γ b for γ < In particular, d T x ≤ δ is equivalent to d T x ≥  δ UCSD Center for Computational Mathematics Slide 10/47, Friday, January 7th, 2011 We cannot draw a picture every time we want to check whether or not a point ¯ x is feasible! Definition The value or residual of a constraint a T x ≥ b at a point ¯ x is r (¯ x ) 4 = a T x b . At an arbitrary point ¯ x r (¯ x ) is ≥ , if a T x ≥ b is satisfied (feasible) at ¯ x = 0 , if a T x ≥ b is active at ¯ x < , if a T x ≥ b is violated (infeasible) at ¯ x UCSD Center for Computational Mathematics Slide 11/47, Friday, January 7th, 2011...
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.
 Spring '08
 staff
 Linear Programming, Geometry

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