Mathematical Programming Homework 01 PDF

MATH 4310/5910R: Mathematical Programming- Homework 01 1. Sketch the regions described by the set X = {x = (x1 , x2 ) ∈ R2 : Ax ≤ b} where A and b are as given below. In each case, state whether these regions are empty or not and whether it is bounded or not. (a)...

MATH 4310/5910R: Mathematical Programming- Homework 01 1. Sketch the regions described by the set X = {x = (x1 , x2 ) ∈ R2 : Ax ≤ b} where A and b are as given below. In each case, state whether these regions are empty or not and whether it is bounded or not. (a)     1 1 4 A = 2 −1 , b = 6 0 1 2 (b)     −1 0 0 0 −1 0 A=  , b=   2 3 6 1 −1 5 (c)     1 1 5 A = −1 −2 , b = −12 −1 0 0 2. Let X = {x = (x1 , x2 ) ∈ R2 : x1 + x2 ≥ 2, x1 ≥ 0, x2 ≥ 0}. Answer the following: (a) How is a direction of X defined? (b) Find all extreme directions of X. (c) Determine all directions of X.   2 3. Let X = x = (x1 , x2 ) ∈ R : −x1 + x2 ≤ 2; −x1 + 2x2 ≤ 6; x1 ≥ 0, x2 ≥ 0. (a) Show that X is a convex set. (b) Find all extreme points and extreme directions of X. 4. Consider the optimization problem. max z = 3x1 + x2 s.t. − x1 + 2x2 ≤ 0 x1 , x2 ≥ 0 (a) Sketch the feasible region in the (x1 , x2 ) space. (b) Solve the problem geometrically. (c) Draw the requirement space and interpret feasibility. 1