Lecture2

# Lecture2 - C&O 355 Mathematical Programming Fall 2010...

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C&O 355 Mathematical Programming Fall 2010 Lecture 2 N. Harvey

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Outline LP definition & some equivalent forms Example in 2D Theorem: LPs have 3 possible outcomes Examples Linear regression, bipartite matching, indep set Solutions at corner points Duality and certificates
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 Note: “<“ and “>” are not allowed in constraints Because we want the feasible region to be closed , in the topological sense.

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

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x 1 x 2 x 2 - x 1 ¸ 1 x 1 + 6x 2 · 15 4x 1 - x 2 ¸ 10 (0,0) Feasible region is empty Constraint 2D Example x 1 ¸ 0 x 2 ¸ 0 Infeasible
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Lecture2 - C&O 355 Mathematical Programming Fall 2010...

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