hw1 - ORIE 321/521 Optimization II Summer 2006 Homework 1...

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ORIE 321/521 Optimization II Summer 2006 Homework # 1 Due: Tuesday, July 18, 4 p.m. in the OR homework drop box for OR321. Please print your name clearly on the first page of your homework. 1. (a) Give brief (but precise) mathematical definitions for: a convex set; an extreme point; a basic feasible solution. (b) Suppose S and T are convex sets with S T and x is a point which is in both S and T . For the following statements either prove valid or demonstrate false by example (a diagram will suffice to show false): (i) If x is an extreme point of S , then x is also an extreme point of T . (ii) If x is an extreme point of T , then x is also an extreme point of S . 2. Use LP duality to prove the Farkas Theorem : 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 . 3. 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) We want to show that any ¯ x F can be expressed as a convex combination of the extreme points of F . First, list the extreme points of
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