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# handout13 - Math 171A Linear Programming Lecture 13...

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Unformatted text preview: Math 171A: Linear Programming Lecture 13 Optimality Conditions for LP Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, February 4th, 2011 Recap: convex sets Definition (Convex set) A set S is convex if, for every x , y ∈ S , it holds that z = (1- θ ) x + θ y ∈ S for all 0 ≤ θ ≤ 1 Note that z = (1- θ ) x + θ y = x + θ ( y- x ) = x + θ p , with p = y- x i.e., steps along any p joining x , y ∈ S give a point in S . UCSD Center for Computational Mathematics Slide 2/24, Friday, February 4th, 2011 Convex set x z = x + θp y = x + p S Result The feasible region F = { x : Ax ≥ b } is either empty or convex. Proof: The result is trivial if F is empty. Assume that F is nonempty, with x , y ∈ F , i.e., x ∈ F ⇒ Ax ≥ b y ∈ F ⇒ Ay ≥ b If θ ∈ [0 , 1] then A ( (1- θ ) x + θ y ) = (1- θ ) Ax + θ Ay ≥ (1- θ ) b + θ b = b Thus, (1- θ ) x + θ y ∈ F . UCSD Center for Computational Mathematics Slide 4/24, Friday, February 4th, 2011 Implication: Given x * ∈ F , we can write every other x ∈ F in the form x = x * + p , with p = x- x * Then Ap = A ( x- x * ) = Ax- Ax * ≥ b- b = 0 ⇒ that Ap ≥ 0 and p is a feasible direction at x * ....
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handout13 - Math 171A Linear Programming Lecture 13...

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