{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW8_Sol

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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

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
Ask a homework question - tutors are online