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handout5 - Recap basic properties of an LP Math 171A Linear...

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Math 171A: Linear Programming Lecture 5 Review of Linear Equations I Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Wednesday, January 12th, 2011 Recap: basic properties of an LP An LP is either infeasible , unbounded or has an optimal solution . An optimal solution always lies on the boundary of the feasible region. Every point on the boundary of the feasible region satisfies a linear system of equations that is either square, underdetermined or overdetermined. UCSD Center for Computational Mathematics Slide 2/40, Wednesday, January 12th, 2011 Review of linear equations We review properties of systems of linear equations Ax = b where A is an m × n matrix and b is an m -vector. We say A R m × n and b R m . We make no assumptions on the shape of A we cannot say that x = A - 1 b , in general. UCSD Center for Computational Mathematics Slide 3/40, Wednesday, January 12th, 2011 Example: Mixing chemotherapy drugs. Set of m typical patient characteristics (e.g., weight, age, etc.): patient 1 , patient 2 , . . . , patient m Set of n drug ingredients: drug 1 , drug 2 , . . . , drug n UCSD Center for Computational Mathematics Slide 4/40, Wednesday, January 12th, 2011
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Suppose that a ij = estimated effect on patient i of drug j x j = quantity of drug j in the mixture then the effect of drug mixture on patient i is a i 1 x 1 + a i 2 x 2 + · · · + a in x n UCSD Center for Computational Mathematics Slide 5/40, Wednesday, January 12th, 2011 Problem: If b i is the desired effect of drug mixture on patient i , find x 1 , x 2 , . . . , x n , such that b i = a i 1 x 1 + a i 2 x 2 + · · · + a in x n for i = 1, 2, . . . , m This is the same as: find x 1 , x 2 , . . . , x n , such that b i = n X j =1 a ij x j for i = 1, 2, . . . , m UCSD Center for Computational Mathematics Slide 6/40, Wednesday, January 12th, 2011 In matrix form: b = Ax , where b = b 1 b 2 . . . b m , A = a 11 a 12 · · · a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 1 · · · a mn , x = x 1 x 2 . . . x n This is a system of linear equations . UCSD Center for Computational Mathematics Slide 7/40, Wednesday, January 12th, 2011 Another notation: write A by columns: A = a 1 a 2 · · · a n | {z } columns of A with a j = a 1 j a 2 j . . . a mj In this case, a j = effects of drug j with a j R m , i.e., Ax = n X j =1 a j x j UCSD Center for Computational Mathematics Slide 8/40, Wednesday, January 12th, 2011
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An obvious question: Is it possible to get the desired effect using any mixture? i.e., does there exist an x such that b = Ax ? UCSD Center for Computational Mathematics Slide 9/40, Wednesday, January 12th, 2011 Definition A system of equations is compatible if there exists a vector x such that b = Ax .
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