mid-2-self-check

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 6 = 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 6 = j . Then 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 6 = 0 in the current slack form also a nonzero value, otherwise a pivot operation would decrease the number of nonzero basic variables.nonzero basic variables....
View Full Document

This document was uploaded on 10/05/2010.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online