Notes - Math 171A LINEAR PROGRAMMING Class Notes c...

Info iconThis preview shows pages 1–5. 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 Class Notes c circlecopyrt 1998. Philip E. Gill, Walter Murray and Margaret H. Wright Department of Mathematics University of California, San Diego, La Jolla, CA 92093-0112. January 2007 Contents 1 Background 7 1.1. Definitions and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.3 Matrices with Special Structure . . . . . . . . . . . . . . . . . . . . 15 1.2. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.1 Linear Dependence and Independence . . . . . . . . . . . . . . . . . 18 1.2.2 Range and Null Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.3 Singular and Nonsingular Matrices . . . . . . . . . . . . . . . . . . . 23 1.3. Solving Rectangular Linear Systems . . . . . . . . . . . . . . . . . . . . . . 25 1.3.1 Full Row Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 Full Column Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.3 Characterization of a Solution . . . . . . . . . . . . . . . . . . . . . 29 2 Linear Programming 31 2.1. Formulating a Linear Program . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.1 The Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.2 Formulation as a Linear Program . . . . . . . . . . . . . . . . . . . 33 2.2. Properties of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 The Normal Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 Level Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.3 One-Dimensional Variation . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.4 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Equality Constraints 41 3.1. Properties of Linear Equality Constraints . . . . . . . . . . . . . . . . . . . 41 3.1.1 Feasible Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.2 Feasible Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2. Optimality for Equality Constraints . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Feasible Descent Directions . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Derivation of Optimality Conditions . . . . . . . . . . . . . . . . . . 45 3 4 CONTENTS January 4, 2007 4 Inequality Constraints 49 4.1. Properties of Linear Inequality Constraints . . . . . . . . . . . . . . . . . . 49 4.1.1 Feasible Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.2 Active and Inactive Constraints . . . . . . . . . . . . . . . . . . . . 52 4.1.3 Feasible Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.1.4 The Step to the Nearest Constraint . . . . . . . . . . . . . . . . . . 57 4.1.5 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Vertices ....
View Full Document

This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

Page1 / 150

Notes - Math 171A LINEAR PROGRAMMING Class Notes c...

This preview shows document pages 1 - 5. Sign up to view the full document.

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