Tutorial 2 - y = a x b which minimizes the sum of absolute...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 2 1. (a) Reformulate the following problem as a linear programming problem: max min( x 1 ,x 2 ) s.t. | 2 x 1 + x 2 | ≤ 7 3 x 1 - x 2 1 + x 1 + x 2 0 . 5 x 1 ,x 2 0 . (b) Consider the problem of minimizing a cost function of the form c 0 x + f ( d 0 · x ), subject to the linear constraints Ax b . Here d is a given vector and the function f : R R is as specified in the following figure. Provide a linear programming formulation of this problem. - 1 2 @ @ @ @ @ @ @ @ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ -1 2 x 2. Consider a LP problem in compact form min c 0 x s.t. Ax b . Suppose the LP problem has k optimal solutions x (1) , x (2) , ··· , x ( k ) . Show that k X i =1 λ i x ( i ) is an optimal solution if k X i =1 λ i = 1 and λ i 0 for i = 1 , 2 , ··· ,k . 3. Consider the polyhedron P = { x R 3 | 2 x 1 - x 2 - x 3 1 ,x i 0 ,i = 1 , 2 , 3 } . (a) Is x 1 = (1 , 1 , 1) 0 a vertex? (b) Is x 2 = (1 , 0 , 1) 0 an extreme point? 1
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4. You are given a set of m data points of the form ( y i , x i ) where y i R and x i R n and need to develop a model to predict the value of y from the knowledge of the vector x . (a) Formulate a linear program to find the best-fit linear model
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Unformatted text preview: y = a x + b which minimizes the sum of absolute deviations of each observed value of y i from it’s predicted value a x i + b . (b) Modify the formulation in part (a) to minimize the maximum absolute devia-tion over all observed values y i from it’s predicted value a x i + b . 5. Consider the polyhedron P = { x ∈ R 2 | x 1 ≥ ,x 2 ≥ , a i x ≥ b i ,i = 1 , 2 , 3 , 4 } (shaded region), where each line L i is defined by a i x = b i , i = 1 , 2 , 3 , 4. For each of the vec-tors represented by points A,B,C,D,E and F , determine (a) if it represents a feasible solution; (b) all active constraints at this point; (c) its rank. Conclude which vectors are basic feasible solutions. x 1 x 2 A O B C D E F L 4 L 3 L 2 L 1 2...
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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.

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Tutorial 2 - y = a x b which minimizes the sum of absolute...

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