mthsc810-lecture04-2x2

# mthsc810-lecture04-2x2 - MthSc 810 Mathematical Programming...

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Unformatted text preview: 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 } affine : { 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 } Extreme points and vertices Assume P ⊆ R n . ◮ x ∈ P is an extreme point of P if ∄ y , z ∈ P \ { x } , λ ∈ [ , 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 nAn = 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 . 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....
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## This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

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mthsc810-lecture04-2x2 - MthSc 810 Mathematical Programming...

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