hw1 - IE426 Problem Set#1 Prof Jeff Linderoth IE 426 –...

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Unformatted text preview: IE426 Problem Set #1 Prof Jeff Linderoth IE 426 – Problem Set #1 Due Date: September 14, 2006. 4:30PM. 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 . 1.2 Problem x ∈ R 2 ,f ( x 1 ,x 2 ) = x 2 1- x 2 2 1.3 Problem x ∈ R 1 ,f ( x ) = x 3 . 1.4 Problem x ∈ R 1 + ,f ( x ) = x 3 . 1.5 Problem x ∈ R 2 ,f ( x 1 ,x 2 ) = 8 x 2 1 + 16 x 2 2- 2 x 1 x 2 1.6 Problem ( x,y,z,w ) ∈ R 4 + ,f ( x,y,z,w ) = 5 x- 8 w + 3 y- 4 z + 77 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 Problem 2 Page 1 IE426 Problem Set #1 Prof Jeff Linderoth 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 ≥ } 2.2 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 ≤ 4 } 2.3 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 ≥ 4 } 2.4 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 = 4 } 2.5 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : x 1 + x 2 = 4 } 2.6 Problem A ∈ R m × n ,b ∈ R m . P = { x ∈ R n : Ax ≤ b,x ≤ } 2.7 Problem P = { ( x 1 ,x 2 ) ∈ R 2 : (3 x 1 + 2 x 2 ) / ( x 1 + x 2 ) = 5 } 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 2 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 ≥ 3.2 Problem max x 1 + x 2 s.t. 3 x 1- x 2 + x 3 ≥ 1 x 1 ,x 2 ,x 3 ≥ 3.3 Problem EXTRA CREDIT min x 1- 3 x 2 + x 3 s.t. 3 x 1- x 2 + x 3 = 1 x 1 ,x 2 ,x 3 ≤ 10 x 1 ≥ x 3 ≥ - 2 3.2 The Joy of Sets 3.4 Problem You are given the following index sets: S def = { 1 , Illinois ,Y } T def = { A,B,C,D } , decision variables x ij ∀ i ∈ S, ∀...
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hw1 - IE426 Problem Set#1 Prof Jeff Linderoth IE 426 –...

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