mthsc810-lecture04-1x2

mthsc810-lecture04-1x2 - MthSc 810: Mathematical...

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MthSc 810: Mathematical Programming Lecture 4 Pietro Belotti Dept. of Mathematical Sciences Clemson University September 6, 2011 Reading for today: Sections 2.3-2.7 Reading for Sep. 8: Sections 2.5-2.7 Hulls Consider two vectors x and x ′′ of R n . The . .. hull of { x , x ′′ } is the set of . .. combinations of x and x ′′ linear : { a x + a ′′ x ′′ , a R , a ′′ R } afFne : { a x + a ′′ x ′′ , a R , a ′′ R : a + a ′′ = 1 } conic : { a x + a ′′ x ′′ , a R + , a ′′ R + } convex : { a x + a ′′ x ′′ , a R + , a ′′ R + : a + a ′′ = 1 }
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Extreme points and vertices Assume P R n . x P is an extreme point of P if y , z P \ { x } , λ [ 0 , 1 ] : x = λ y + ( 1 λ ) z x P is a vertex of P if c R n : c x < c y y P \ { x } Consider P = { x R n : A x b } and a vector x R n . An inequality a x b is binding at x if a x = b Straight from your Algebra book A system A x = b can be rewritten as x 1 A 1 + x 2 A 2 + . . . + x n A n = b where A i are the columns of A . Therefore, looking for a solution to A x = b is akin to writing b as a combination of A 1 , A 2 . . . , A n .
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Solutions and binding constraints Theorem If a system of m inequalities in n variables a 1 x b 1 a 2 x b 2 · · · a m x b m has a subset I of binding inequalities at x , i.e. a
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mthsc810-lecture04-1x2 - MthSc 810: Mathematical...

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