Chapter 1 Introduction q Linear Programming 2019 1

Chapter 1. Introduction q Linear Programming 2019 1

q Linear Programming 2019 2

q Linear Programming 2019 3

Important submatrix multiplications q Linear Programming 2019 4

q Linear Programming 2019 5

Standard form problems q Linear Programming 2019 6

q Linear Programming 2019 7

1. 2 Formulation examples q Linear Programming 2019 8

q Linear Programming 2019 9

q Linear Programming 2019 10

q Linear Programming 2019 11

q path based formulation has smaller number of constraints, but enormous number of variables. can be solved easily by column generation technique (later). Integer version is more difficult to solve. q Extensions: Network design - also determine the number and type of facilities to be installed on the links (and/or nodes) together with routing of traffic. q Variations: Integer flow. Bifurcation of traffic may not be allowed. Determine capacities and routing considering rerouting of traffic in case of network failure, Robust network design (data uncertainty), . . . Linear Programming 2019 12

q Linear Programming 2019 13

q Linear Programming 2019 14

q Linear Programming 2019 15

q Variations Ø What if there are many choices of hyperplanes? any reasonable criteria? Ø What if there is no hyperplane separating the two classes? Ø Do we have to use only one hyperplane? Ø Use of nonlinear function possible? How to solve them? • SVM (support vector machine), convex optimization Ø More than two classes? Linear Programming 2019 16

1. 3 Piecewise linear convex objective functions q Linear Programming 2019 17

q x ( 1 = 1) y ( 1 = 0) Linear Programming 2019 18

q Linear Programming 2019 19

Picture of convex function Linear Programming 2019 20

q Linear Programming 2019 21

q Linear Programming 2019 22

q Linear Programming 2019 23

q Min of piecewise linear convex functions Linear Programming 2019 24

q Q: What can we do about finding maximum of a piecewise linear convex function? maximum of a piecewise linear concave function (can be obtained as minimum of affine functions)? Minimum of a piecewise linear concave function? Linear Programming 2019 25

q Linear Programming 2019 26

q Linear Programming 2019 27

Problems involving absolute values q Linear Programming 2019 28

Data Fitting q Linear Programming 2019 29

q Linear Programming 2019 30

Approximation of nonlinear function. 0 Linear Programming 2019 31

Linear Programming 2019 32

1. 4 Graphical representation and solution q Linear Programming 2019 33

q Geometry in 2 -D 0 Linear Programming 2019 34

q 0 Linear Programming 2019 35

q Linear Programming 2019 36

q q See text sec. 1. 5, 1. 6 for more backgrounds Linear Programming 2019 37