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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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This note was uploaded on 02/29/2008 for the course IE 426 taught by Professor Linderoth during the Spring '08 term at Lehigh University .
 Spring '08
 Linderoth

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