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handout3 - Math 171A: Linear Programming Lecture 3 Geometry...

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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 ( a ) T x b for > a T x b is equivalent to ( a ) T x b for < In particular, d T x is equivalent to- d T x - 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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handout3 - Math 171A: Linear Programming Lecture 3 Geometry...

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