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Unformatted text preview: Tutorial 3: Outline of Solutions Q1. (a) For each of the following sets, decide whether it is representable as a polyhedron? (i) The set of all ( x,y ) ∈ R 2 satisfying the constraints x cos θ + y sin θ ≤ 1 ∀ θ ∈ [0 ,π/ 2] x ≥ y ≥ . (ii) The set of all x ∈ R satisfying the constraint x 2 8 x + 15 ≤ 0. Solution : x y (1,0) (0,0) (0,1) x 3 5 The first set is a quarter circle in the positive orthant of radius 1. Since an infinite number of linear constraints are needed to describe (i), it is not a polyhedron. The second set can be expressed as x ∈ R satisfying the two linear constraints x ≥ 3 , x ≤ 5. Hence it is representable as a polyhedron. (b) Let f : R n → R be a convex function and let a be some constant. Show that the level set S = { x ∈ R n  f ( x ) ≤ a } is a convex set. Solution : Given x , y ∈ S , we see that for λ ∈ [0 , 1] f ( λ x + (1 λ ) y ) ≤ λf ( x ) + (1 λ ) f ( y ) (From definition of convex function) ≤ λa + (1 λ ) a (Since x , y ∈ S ) = a....
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.
 Spring '10
 TanBanPin
 Sets

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