171A: Math Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2011
Homework Assignment #4 Due Friday February 4, 2011 Starred exercises require the use of Matlab. Exercise 4.1. Suppose that the constant vector c is such that cTp 0 for all p such that Ap = 0. Show that this implies that cTp = 0 for all p such that Ap = 0. Exercise 4.2. Consider the equality-constrained linear program: minimize cTx subject to Ax = b, where -8 -2 -1 5 0 1 1 4 A= , b= and c = 6 . 3 -1 1 4 2 -5 3 5 (a) Find a feasible point x for this problem. Is it optimal? (b) Find unique vectors cN null(A) and cR range(AT ) such that c = cR + cN . Hence find a feasible direction p such that (x + p) < (x) for all positive . Exercise 4.3. Consider the equality-constrained linear program of minimizing cTx subject to Ax = b. For each of the following instances of A, b and c, determine whether or not the linear program has a solution, If not, explain why not. If so, find a minimizer x and a Lagrange multiplier , and explain whether or not they are unique. 8 -4 2 2 10 1 2 , b = 2 and c = 0 . (a) A= 3 -1 3 1 1 5 2 17 -4 1 3 2 -1 (b) A= , b= and c = -1 . -3 -8 2 6 3 -10 1 9 14 5 (c) A = 3 7 , b = 2 and c = . 3 1 -2 1 (d) The same matrix A and vector b as in part (b), and 1 1 c = . 1 1
2
Mathematics 171A
(e) The same matrix A and vector c as in part (c), and 6 b = 3 . 1 Exercise 4.4. Consider the equality-constrained linear program: minimize cTx subject to Ax = b. If this problem has a bounded minimum , show the following: (i) = cTx = cTx for any two feasible points x and x; T T (ii) = b = b for any two sets of Lagrange multipliers and . Exercise 4.5. Consider the equality-constrained linear program: minimize cTx subject to Ax = b. Show that if A has full column rank and b range(A), then this problem always has a unique bounded solution, regardless of the definition of c. Do Lagrange multipliers exist? If they exist, are they unique? Exercise 4.6. Let F denote the feasible region F = {x Rn : Ax = b}, where A be an m n matrix of rank r and b is an m vector. Assume that there is at least one feasible point. Let Ar denote an r n matrix of r linearly independent rows of A and let br denote the corresponding r-vector of right-hand sides. Show that F is identical to the = set Fr {x Rn : Ar x = br }. This problem should be a good test of your linear algebra. Don't worry, it's too tricky ever to be on a test!