1. Linear Programming 2012 1 Chapter 1 Introduction

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4. 4 Important submatrix multiplications

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6. 6 Standard form problems

7. Linear Programming 2012 7  Any (practical) algorithm can solve the LP problem in equality form only (except nonnegativity)  Modified form of the simplex method can solve the problem with free variables directly (without using difference of two variables). It gives more sensible interpretation of the behavior of the algorithm.

8. Linear Programming 2012 8 1.2 Formulation examples

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12. Linear Programming 2012 12  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.  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.  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),...

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16.  Variations  What if there are many choices of hyperplanes? any reasonable criteria?  What if there is no hyperplane separating the two classes?  Do we have to use only one hyperplane?  Use of nonlinear function possible? How to solve them? SVM (support vector machine), convex optimization  More than two classes? Linear Programming 2012 16

17. Linear Programming 2012 17 1.3 Piecewise linear convex objective functions

18. Linear Programming 2012 18 x ( 1 = 1) y ( 1 = 0)

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20. Linear Programming 2012 20 Picture of convex function

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24. Linear Programming 2012 24  Min of piecewise linear convex functions

25. Linear Programming 2012 25  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?

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28. Linear Programming 2012 28 Problems involving absolute values

29. Linear Programming 2012 29 Data Fitting

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31. Linear Programming 2012 31 0 Approximation of nonlinear function.

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33. Linear Programming 2012 33 1.4 Graphical representation and solution

34. Linear Programming 2012 34  Geometry in 2-D 0

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37. Linear Programming 2012 37  See text sec. 1.5, 1.6 for more backgrounds