homework-8-solutions

homework-8-solutions - MthSc810 Mathematical Programming...

Info iconThis preview shows pages 1–3. 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: MthSc810 Mathematical Programming Fall 2011, solutions to Homework #8. Problem 1. Write the dual of the following problems: 1. The Klee-Minty example for general n ; 2. The bank loan model seen in lecture 2, slide 21; 3. The production planning model (same lecture, slide 11). Solutions. 1. Primal: min- n j =1 10 n j x j s.t. x i + 2 i 1 j =1 10 i j x j 100 i 1 i = 1 , 2 . . . , n x i i = 1 , 2 . . . , n The dual has n constraints and n variables, just like the primal: max n i =1 100 i 1 u j s.t. u i + 2 n j = i +1 10 j i u j - 10 n i i = 1 , 2 . . . , n u i i = 1 , 2 . . . , n. 2. Primal: min y 1 + y 2 s.t. y 1 . 05 x 1 y 1 5 + 0 . 08( x 1- 100) y 2 . 03 x 2 y 2 4 . 2 + 0 . 12( x 2- 140) x 1 + x 2 = 300 x 1 x 2 . Lets rewrite it so that variables are on the left-hand side, and constants on the rhs (and write the dual variables names next to each primal constraint): min y 1 + y 2 s.t.- . 05 x 1 + y 1 ( u 1 )- . 08 x 1 + y 1 - 3 ( u 2 )- . 03 x 2 + y 2 ( u 3 )- . 12 x 2 + y 2 - 12 . 6 ( u 4 ) x 1 + x 2 = 300 ( u 5 ) x 1 , x 2 . Dual: max- 3 u 2- 12 . 6 u 4 + 300 u 5 s.t.- . 05 u 1- . 08 u 2 + u 5 - . 03 u 3- . 12 u 4 + u 5 u 1 + u 2 = 1 u 3 + u 4 = 1 u 1 , u 2 , u 3 , u 4 . 3. Primal: min 12 i =1 c i x i x i + y i 1 = d i + y i i = 1 , 2 . . . , 12 x i P i = 1 , 2 . . . , 12 y i i = 1 , 2 . . . , 12 y = y 12 = 0 Rewrite it to ease writing the dual. Because the two non-trivial constraints are written for various values of i , the corresponding dual will be indexed similarly. However, the two trivial constraints y = 0 and y 12 = 0 will be associated with dual variables rather than used for simplifying the problem (which would be an equally good solution)....
View Full Document

This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

Page1 / 6

homework-8-solutions - MthSc810 Mathematical Programming...

This preview shows document pages 1 - 3. Sign up to view the full document.

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