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mid-2-self-check

# mid-2-self-check - Spring 2010 CS530 Analysis of Algorithms...

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Spring 2010 CS530 – Analysis of Algorithms S ELF - CHECK QUESTIONS 2 I may keep adding to this list, please come back checking... Problem 1. Let A be a positive semidefinite matrix. (a) Show that a ii 0 for all i . Solution. Consider a vector x with x i = 1 and x j = 0 for all j = i . Now, a ii = x T Ax 0. (b) Show that if a ii = 0 then a i j = a ji = 0 for all j . Solution. Consider a vector x with x i = 1 and x k = 0 for k = j . Then 0 x T Ax = a ii + 2 a i j x j + a j j x 2 j = 2 a i j x j + a j j x 2 j . The minimum of the right-hand side is 0 only if a i j = 0. Problem 2. How do you recognize it in the simplex algorithm that an optimal solution has been reached? Solution. Each nonbasic variable enters the objective function with a non- positive coefficient. Problem 3. Prove that if a linear program has a nonbasic optimal solution then it has infinitely many optimal solutions. Solution. Let x be a nonbasic optimal solution. For this solution, let us choose the basis in which the number of nonzero nonbasic variables is as small as possible. Let x j be a nonbasic variable whose value is not 0. Then any basic variable x i for which a i j = 0 in the current slack form also a nonzero value, otherwise a pivot operation would decrease the number of nonzero basic variables.

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mid-2-self-check - Spring 2010 CS530 Analysis of Algorithms...

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