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

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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
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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
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