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Unformatted text preview: IE426 Problem Set #1 Prof Jeff Linderoth IE 426 – Problem Set #1 Answers 1 Convex Functions Categorize each of the functions below as convex , concave , or nonconvex over its domain. Note: Be sure to consider whether or not each function is in more than one category. 1.1 Problem x ∈ R 1 ,f ( x ) = x 4 . Answer: f 00 ( x ) = 12 x 2 This is ≥ ∀ x ∈ R 1 . Therefore, f is Convex . ♦ 1.2 Problem x ∈ R 2 ,f ( x 1 ,x 2 ) = x 2 1 x 2 2 Answer: ∇ 2 f ( x ) = 2 2 . f is nonconvex . ♦ 1.3 Problem x ∈ R 1 ,f ( x ) = x 3 . Answer: f 00 ( x ) = 6 x . For x ≥ ,f 00 ( x ) ≥ , for x ≤ ,f 00 ( x ) ≤ . Therefore f is nonconvex on the domain R . ♦ 1.4 Problem x ∈ R 1 + ,f ( x ) = x 3 . Answer: f 00 ( x ) = 6 x . For x ≥ ,f 00 ( x ) ≥ , so f is convex on the domain R + . ♦ 1.5 Problem x ∈ R 2 ,f ( x 1 ,x 2 ) = 8 x 2 1 + 16 x 2 2 2 x 1 x 2 Problem 1 Page 1 IE426 Problem Set #1 Prof Jeff Linderoth Answer: Answer: ∇ 2 f ( x ) = 16 2 2 32 . Since 16 > , 32 > , and 16(32) ( 2)( 2) > , f is convex. ♦ 1.6 Problem ( x,y,z,w ) ∈ R 4 + ,f ( x,y,z,w ) = 5 x 8 w + 3 y 4 z + 77 Answer: This is a linear function. Linear functions are both convex and concave . ♦ 1.7 Problem EXTRA CREDIT x ∈ R 3 ,f ( x 1 ,x 2 ,x 3 ) = x 2 1 3 x 2 2 2 x 3 3 4 x 1 x 2 + 2 x 1 x 3 + 4 x 2 x 3 2 Convex Sets Categorize each of the sets P below as convex or nonconvex. If anyone writes “concave” I will bludgeon them. 2.1 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : 7 x 1 2 x 2 ≤ 8 ,x 1 ≥ } Answer: P is a polyhedron. Polyhedra are convex ♦ 2.2 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 ≤ 4 } Answer: P is the area inside a circle. This is a convex region. ♦ 2.3 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 ≥ 4 } Answer: P is the area outside a circle. This is a nonconvex region. ♦ Problem 2 Page 2 IE426 Problem Set #1 Prof Jeff Linderoth 2.4 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 = 4 } Answer: P is set of points lying exactly on a circle. This is a nonconvex region. ♦ 2.5 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 1 + x 2 = 4 } Answer: P is a polyhedron. Polyhedra are convex ♦ 2.6 Problem A ∈ R m × n ,b ∈ R m . P = { x ∈ R n : Ax ≤ b,x ≤ } Answer: P is a polyhedron. Polyhedra are convex ♦ 2.7 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : (3 x 1 + 2 x 2 ) / ( x 1 + x 2 ) = 5 } Answer: Trick Question. It seems like P is the line x 2 = (2 / 3) x 1 , however the point (0 , 0) is not in the set P , so if you choose one point on either side, the line segment is not entirely inside, so the set is nonconvex . ♦ 3 Math Review 3.1 Canonical Form Put the following linear programs into the canonical form: max x ∈ R n c T x subject to Ax ≤ b x ≥ Problem 3 Page 3 IE426 Problem Set #1 Prof Jeff Linderoth 3.1 Problem min x 1 2 x 2 + 7 x 3 s.t. 8 x 1 + x 2 = 0 x 3 ≥ 4 x 1 ,x 2 ,x 3 ≥ Answer: max x 1 + 2 x 2 7 x 3 s.t. 8 x 1 + x 2 ≤ 8 x 1 x 2 ≤ x 3 ≤  4 x 1 ,x 2 ,x 3 ≥ ♦ 3.2 Problem max x 1 + x 2...
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 Spring '08
 Linderoth
 Linear Programming, Convex set, Jeff Linderoth, Prof Jeff Linderoth

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