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# hw3 - Let(P be a linear program and let(D be its dual Let x...

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Assignment 3 Due: October 6th, 2005 (Thu), 3:00PM From the notes do the following exercises 4.8.7 (page 56), Exercise 1: Suppose we are given m di±erent numbers { a 1 , . . . , a m } . Con- sider the following linear program (P), which has one variable only, min x subject to x a i ( i = 1 , . . . , m ) 1. Find the dual (D) of (P), 2. Write the complementary slackness conditions for (P) and (D), 3. Suppose y * is an optimal solution to (D), prove using complementary slackness theory that, for i = 1 , . . . , m , y * i = b 1 if a i is the largest number among { a 1 , . . . , a m } 0 otherwise Exercise 2:
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Unformatted text preview: Let (P) be a linear program and let (D) be its dual. Let x * be a feasible solution for (P) and let y * be a feasible solution for (D). Show that if all variables of (P) and (D) are free (unrestricted) then x * is an optimal solution to (P) and y * is an optimal solution to (D). Exercise 3: Prove Theorem 4.9 for the case where (P) is in standard in-equality form. Important: your proof should be self contained. In partic-ular you should NOT convert (P) to a problem in standard equality form and apply Theorem 4.7. 1...
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