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hw3_07

# hw3_07 - MS&E 211 Linear and Nonlinear Optimization Prof...

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MS&E 211 Fall 2007 Linear and Nonlinear Optimization Oct 18, 2007 Prof. Yinyu Ye Homework Assignment 3: Due 3:15pm Friday, Oct 26 There is a homework collecting box outside of professor Ye’s office (Terman 316) for you to submit your homework. No late homework accepted! Reading. Read sections 6.1 to 6.7 in the text book. Problem 1 : Sensitivity Analysis 1 Consider the following linear programming problem Maximize 9 x 1 + 8 x 2 + 5 x 3 Subject to 2 x 1 + 3 x 2 + x 3 4 5 x 1 + 4 x 2 + 3 x 3 11 x 1 , x 2 , x 3 0 Let x 4 and x 5 denote the slack variables for the respective functional constraints. First, we convert the problem into a minimization problem, then we apply the simplex method. The final simplex tableau is 0 2 0 2 1 19 x 1 1 5 0 3 -1 1 x 3 0 -7 1 -5 2 2 a) Write the dual problem. b) Will the optimal basis change if we change the RHS vector b from (4;11) to (6;11). c) Will the optimal basis change if the coefficient of x 2 in the objective function is changed from 8 to 11. d) Suppose that the following constraint is added to the problem: x 1 + x 2 + x 3 6 . Using sensitivity, find the new optimal solution. 1 This is a revised version of problem 6.7-18 in the textbook

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Problem 2 : Asset Allocation Duality Your rich uncle Warren has been impressed with your performance at Stanford and has decided that he’d like you in invest \$10M for him. Warren has a lot of friends currently starting new companies that need money. He’d like you to invest his \$10M among these companies such that you maximize the total return. There are a fixed number of companies in which you can choose to invest your assets. Each company requires some fixed amount of capital and has a promised return. For each company, you must decide how much of its
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hw3_07 - MS&E 211 Linear and Nonlinear Optimization Prof...

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