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Unformatted text preview: CO350 Linear Programming Chapter 5: Basic Solutions 27th May 2005 Chapter 5: Basic Solutions 1 Notation ( P ) maximize c T x subject to Ax = b x ≥ Many times, we will assume that A has rank = # rows. This is w.l.o.g.: Apply Gaussian Elimination to [ A  b ] and either ? conclude Ax = b has no solution, or ? eliminate redundant row to get A x = b where A has rank = # rows. Let A j denote column j of the matrix A A B denote the submatrix [ A j : j ∈ B ] of the matrix A (Defn ) Basis (Pg 59) (Do NOT confused with basis of vector space) A subset B of { 1 , 2 ,...,n } such that (a)  B  = m (i.e., B has m elements), and (b) A B is nonsingular (i.e., invertible). Note that B is a basis of A if and only if columns of A B forms a basis of the vector space R m . Chapter 5: Basic Solutions 2 Example (NOT in notes) A = 2 6 6 6 4 2 0 4 2 0 1 1 1 1 1 2 3 7 7 7 5 B = { 1 , 2 , 3 } is a basis as A B = 2 6 6 6 4 2 0 4 0 1 1 1 1 3 7 7 7 5 is nonsingular. B = { 1 , 3 , 4 } is a basis as A B = 2 6 6 6 4 2 4 2 1 1 1 2 3 7 7 7 5 is nonsingular. B = { 1 , 2 , 4 } is NOT a basis as A B = 2 6 6 6 4 2 0 2 0 1 1 1 1 2 3 7 7 7 5 is singular. Chapter 5: Basic Solutions 3 Suppose B is a basis for A . Consider the system of n equations in n unknowns. Ax = b x j = 0 ( j / ∈ B ) This system has a unique solution. (Why?) (Defn ) Basic solution determined by a basis B The solution to the above system of equations. (Defn ) Basic solution of Ax = b The basic solution determined by some basis B . Note: A basic solution always have at least n m zeros since x j = 0 ( j / ∈ B ) I.e., a basic solution always have at most m nonzeros. Example (NOT in notes) Consider Ax = b , where A = 2 6 6 6 4 2 0 4 2 0 1 1 1 1 1 2 3 7 7 7 5 b = 2 6 6 6 4 2 1 5 3 7 7 7 5 The basic solution determined by B = { 1 , 2 , 3 } is [3 , 1 , 1 , 0] T . The basic solution determined by B = { 1 , 3 , 4 } is [2 , , 1 , 1] T . Chapter 5: Basic Solutions 4 Question (Similar to question on Pg 61) A = 2 4 2 4 4 2 4 0 2 1 1 3 5 b = 2 4 2 1 3 5 Which of the following are basic solutions of Ax = b ?...
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 Fall '97
 Wolkowicz
 Linear Algebra, Vector Space, Convex set, Basic Solutions, basic solution

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