handout5

handout5 - Math 171A: Linear Programming Lecture 5 Review...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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 An obvious question: Is it possible to get the desired effect using any mixture?...
View Full Document

This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.

Page1 / 10

handout5 - Math 171A: Linear Programming Lecture 5 Review...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online