# hw1 - ORIE 3310/5310 Optimization II Summer 2009 Homework 1...

This preview shows pages 1–3. Sign up to view the full content.

ORIE 3310/5310 Optimization II Summer 2009 Homework # 1 Due: Monday, July 6 at 2:30 in the OR homework drop box for OR3310. 1. Recall the Farkas Theorem from linear programming: For any m × n real matrix A and m -vector b exactly one of the two following systems is feasible: (I) Ax = b , x 0 (II) yA 0 , yb < 0 . The LP feasible region F = { x : Ax = b, x 0 } 6 = is bounded provided F ⊆ { x : - δ x j δ j } for some positive constant δ ; i.e., we can place the LP feasible region in a hypercube of side length 2 δ . Suppose that F is such a bounded feasible region. (a) Show that the LP max x F cx has a finite optimum solution value for any objective function c . (b) In our discussion of LP decomposition, we use the fact that any ¯ x F can be expressed as a convex combination of the extreme points of F . In order to establish this result, we first list the extreme points of F as the columns of matrix M . ( F can have only finitely many extreme points. Why? Careful...be precise.) Next, consider the following linear system expressing ¯ x as a convex combination of the extreme points of F : { = ¯ x, X j λ j = 1 , λ j 0 j } . Use the Farkas Theorem to show that this linear system must have a solution for any ¯ x F . (c) What about the converse of the result in ( b ) ? I.e., must any convex combination of extreme points of F be in F ? You should justify your answer. 1

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

View Full Document
2. Two different types of coal, X and Y , are mined at 5 locations and used to produce electrical power in 4 power plants. The daily rates of production at the mines and the daily demands at the power plants are given in the following tables (units are hundreds of tons): 1 2 3 4 5 X 5 20 10 0 0 Y 0 0 10 15 15 A B C D X 10 * 15 0 Y 0 * 5 20 The * for plant B refers to the fact that at B either type X or Y can be used, though X is used less effectively – 1.2 tons of X are equivalent to 1.0 tons of Y at plant B . The total equivalent tonnage requirement of type X at plant B is 25 (hundred) tons. Furthermore, due to shipping capacity limitations, the total coal (of types X and Y ) shipped from mine 3 to plant B must not exceed 8 (hundred) tons. The cost of shipping 100 tons of coal from each mine site to each power plant
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern