HW8_Sol - E160 Operations Research I Fall 2008 Homework #8...

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

View Full Document Right Arrow Icon
E160 Operations Research I Fall 2008 Homework #8 Solution Chapter 12 Section 12.9 2. K-T conditions yield (1) 1 - 2x 1 λ 1 = 0 (x 1 constraint) (2) -1 -2x 2 λ 1 = 0 (x 2 constraint) (3) λ 1 (1 - x 1 2 - x 2 2 ) = 0 , λ 1 0 Trying λ 1 >0, we find from (1) and (2) that x 1 = 1/2λ 1 and __ x 2 = -1/2λ 1 . Then (3) yields 1/2λ 1 2 = 1 or λ 1 = \/2 /2. __ __ Finally we find that x 1 = \2/2 and x 2 = -\/2/2. Since the constraint is convex and the objective function is linear, we know that we have found an optimal solution to the problem. 6. Replace the constraint -x 1 + x 2 = 1 by -x 1 + x 2 1 and x 1 - x 2 -1. Then (29)-(33)' yield the following K-T conditions. (1) 2(x 1 - 1) - λ 1 + λ 2 + λ 3 0 (x 1 constraint) (2) 2(x 2 - 2) +λ 1 - λ 2 + λ 3 0 (x 2 constraint) (3) λ 1 (1 + x 1 - x 2 ) = 0 (4) λ 2 (-1 - x 1 + x 2 ) = 0 (5) λ 3 (2 - x 1 - x 2 ) = 0 (6) x 1 (2(x 1 - 1) - λ 1 + λ 2 + λ 3 ) = 0 (7) x 2 (2(x 2 - 2) + λ 1 - λ 2 + λ 3 ) = 0 Try λ 3 >0. Then (5) yields x
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/18/2008 for the course IEOR 41033 taught by Professor Adler during the Spring '08 term at University of California, Berkeley.

Page1 / 2

HW8_Sol - E160 Operations Research I Fall 2008 Homework #8...

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