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Lecture2

# Lecture2 - C&O 355 Lecture 2 N Harvey...

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Lecture 2 N. Harvey http://www.math.uwaterloo.ca/~harvey/

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Outline Example in 2D Possible outcomes Examples Linear regression, bipartite matching, indep set Feasible Region, Convex Sets Local-Search Algorithm
Linear Program General definition Parameters: c, a 1 ,…,a m 2 R n , b 1 ,…,b m 2 R Variables: x 2 R n Terminology Feasible point: any x satisfying constraints Optimal point: any feasible x that minimizes obj. func Optimal value: value of obj. func for any optimal point Objective function Constraints

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Matrix form Parameters: c 2 R n , A 2 R m £ n , b 2 R m Variables: x 2 R n Linear Program General definition Parameters: c, a 1 ,…, a m 2 R n , b 1 ,…, b m 2 R Variables: x 2 R n
Simple LP Manipulations “max” instead of “min” max c T x ´ min c T x ¸ ” instead of “ · a T x ¸ b , -a T x · -b “=” instead of “ · a T x=b , a T x · b and a T x ¸ b

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x 1 x 2 x 2 - x 1 · 1 x 1 + 6x 2 · 15 4x 1 - x 2 · 10 (1,1) (0,0) Feasible region Constraint Optimal point 2D Example (Textbook, Ch 1) x 1 ¸ 0 x 2 ¸ 0 Unique optimal solution exists (3,2)
x 1 x 2 x 2 - x 1 · 1 x 1 + 6x 2 · 15 4x 1 - x 2 · 10 (1/6,1) (0,0) Feasible region Constraint Optimal points 2D Example x 1 ¸ 0 x 2 ¸ 0 Optimal solutions exist: Infinitely many!

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1 x 2 x 2 - x 1 ¸ 1 x 1 + 6x 2 · 15 4x 1 - x 2 ¸ 10
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Lecture2 - C&O 355 Lecture 2 N Harvey...

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