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IOE510+Lecture+02+03+10+Wed

# IOE510+Lecture+02+03+10+Wed - IOE 510 Linear Programming I...

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1 IOE 510 Linear Programming I Wednesday February 3, 2010 Professor Amy Cohn

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2 Announcements Reminder of paper- instead of electronic- homework submission
3 Re-cap from Last Time Constructing Existence of extreme points What is a “line”?

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4 Constructing Extreme Points in Standard Form
5 Polyhedra in Standard Form We will focus on polyhedra of the form: Min cx St Ax = b x > 0 The equations Ax = b defines an affine subspace (loosely, a plane) The polyhedron is the intersection of this affine subspace with the positive orthant

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6 Polyhedra in Standard Form A basic feasible solution must have n lienarly independent active constraints The m equality constraints must be satisfied This leaves n – m more constraints Our only choice is the non-negativity constraints So the basic feasible solutions are at the intersection of the constraints x i = 0
7 Informal Idea To find a basic feasible solution, choose n - m variables and set them to zero This leaves m equations and m unknowns If these equations are not linearly independent, then this is not a basis – disregard If these equations are linearly independent, then they yield a basis In particular, they yield a unique solution in which n constraints are active; we call this a basic solution We call the variables set to zero non-basic and the remaining variables basic If all of the basic variables are non-negative, then it is also a basic feasible solution

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8 Existence of Extreme Points
9 Lines and Extreme Points Claim: Given a non-empty polyhedron in the form Ax > b, the following are equivalent: a) P has at least one extreme point b) P does not contain a line c) There exist n vectors {a i } corresponding to constraints in P that are linearly independent Proof: Lemma 1) b implies a Lemma 2) c implies b Lemma 3) a implies c

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10 Lines and Extreme Points Lemma 1) No line implies at least one EP Proof by construction: Let x be any point in P Case 1: x is an extreme point – complete
11 Lines and Extreme Points Proof (cont.): Case 2: x is not an extreme point

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IOE510+Lecture+02+03+10+Wed - IOE 510 Linear Programming I...

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