Homework 2 Solutions

Homework 2 Solutions - Math 171A: Mathematical Programming...

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Unformatted text preview: Math 171A: Mathematical Programming Instructor: Philip E. Gill Winter Quarter 2009 Homework Assignment #2 Due Friday January 23, 2009 The starred exercises require the use of Matlab . Remember that it is necessary to do all the Matlab assignments to obtain credit for the class. Exercise 2.1. (a) Write the constraints x 1 + x 2 4, x 1 + 3 x 2 6, 6 x 1- x 2 18, 3 x 2 6 and x 1 - 1 in matrix form Ax b . After rearranging some of the constraints we find that A = 1 1 1 3- 6 1 1- 1 1 and b = 4 6- 18 3- 6- 1 . (b) Draw the graphical representation of the feasible region defined by the constraints of part (a) . Mark the infeasible side of the half-space defined by each constraint. Below is the graph with the feasible region. The vertices are A = (- 1 , 5) , B = (1 , 3) , C = (3 . 5 , 3) , D = (4 , 6) , E = (- 1 , 6) 1 2 3 4 5 6-1 1 2 3 4 5 feasible region A E B C D x 1 x 2 2 Mathematics 171A (c) Consider the linear function ( x ) = c T x where c T = (2- 3). On your graph from part (a) , show the halfspace of descent directions emanating from the point x = (2 , 4). The linear function ( x ) =- 8 = c T x is shown in the graph below. The halfspace of descent directions emanating from x is the space above this line. 1 2 3 4 5 6-1 1 2 3 4 5 feasible region A E B C D x 1 x 2 c x (d) Use the graphical method to minimize 2 x 1- 3 x 2 subject to the constraints of part (a) . Give the optimal values of x 1 , x 2 and the minimum value of the objective function. If the objective level curve is shifted along the negative normal of ( x ) we find that the solution x * is located at E = (- 1 , 6) , which lies at the intersection of the hyperplanes defining the constraints x 1 - 1 and x 2 6 . The minimum value of the objective is 2(- 1)- 3(6) =- 20 . (e) Compute the residual vector r ( x ) for the constraints at the point x and find the con- straints whose residuals would decrease after a positive step along the direction p = 1- 2 emanating from x ....
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Homework 2 Solutions - Math 171A: Mathematical Programming...

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