C&O 355
Mathematical Programming
Fall 2010
Lecture 18
N. Harvey
Topics
Network Flow
Max Flow / Min Cut Theorem
Total Unimodularity
Directed Graphs & Incidence Matrices
Proof of Max Flow / Min Cut Theorem
Network Flow
Let D=(N,A) be a directed graph.

C&O 355
Mathematical Programming
Fall 2010
Lecture 16
N. Harvey
Topics
Semidefinite Programs (SDP)
Vector Programs (VP)
Quadratic Integer Programs (QIP)
QIP & SDP for Max Cut
Finding a cut from the SDP solution
Analyzing the cut
The Max Cut Problem
Our fi

C&O 355
Mathematical Programming
Fall 2010
Lecture 15
N. Harvey
Topics
Minimizing over a convex set:
Necessary & Sufficient Conditions
(Mini)-KKT Theorem
Minimizing over a polyhedral set:
Necessary & Sufficient Conditions
Smallest Enclosing Ball Problem

C&O 355: Mathematical Programming
Fall 2010
Lecture 12 Notes
Nicholas Harvey
http:/www.math.uwaterloo.ca/~harvey/
1
Zero-Sum Games
Let M be any m n real matrix, which we use as the payo matrix for a two-player, zero-sum game.
Von Neumanns theorem states t

C&O 355
Mathematical Programming
Fall 2010
Lecture 12
N. Harvey
Daniel Kleitman
First person with finite
Erdos-Bacon number?
What movie is this?
What is the location for this scene?
Who are the people in this scene?
Two-Player Zero-sum Games
Alice and

C&O 355: Mathematical Programming
Fall 2010
Lecture 11 Notes
Nicholas Harvey
http:/www.math.uwaterloo.ca/~harvey/
1
Faces of Polyhedra
Recall from Lecture 10 the following denition.
Denition 1.1. Let P Rn be a polyhedron. A face of P is any set F of the f

C&O 355
Mathematical Programming
Fall 2010
Lecture 10
N. Harvey
What is a corner point?
How should we define corner points?
Under any reasonable definition, point x
should be considered a corner point
x
What is a corner point?
Attempt #1: x is the farth

C&O 355
Mathematical Programming
Fall 2010
Lecture 9
N. Harvey
Topics
Semi-Definite Programs (SDP)
Solving SDPs by the Ellipsoid Method
Finding vectors with constrained distances
LP is great, but
Some problems cannot be handled by LPs
Example: Find v

C&O 355
Mathematical Programming
Fall 2010
Lecture 8
N. Harvey
Polynomial Time Algorithms
P = cfw_ computational problems that can be solved efficiently
i.e., solved in time nc, for some constant c, where n=input size
This is a bit vague
Consider an L

C&O 355, Fall 2010
Lecture 7 Notes
Nicholas Harvey
http:/www.math.uwaterloo.ca/~harvey/
1
Covering Hemispheres by Ellipsoids
Recall our notation B = cfw_ x : x 1 and Hu = x : xT u 0 , where u is an arbitrary unit vector.
The next theorem denes an ellipso

C&O 355
Mathematical Programming
Fall 2010
Lecture 7
N. Harvey
Covering Hemispheres by Ellipsoids
B
u
Let B = cfw_ unit ball .
Let Hu = cfw_ x : xTu0 , where kuk=1.
Find a small ellipsoid B that covers BH.
Rank-1 Updates
Def: Let z be a column vector

C&O 355
Mathematical Programming
Fall 2010
Lecture 6
N. Harvey
Polyhedra
Definition: For any a2Rn, b2R, define
Hyperplane
Halfspaces
Def: Intersection of finitely many halfspaces is a polyhedron
Def: A bounded polyhedron is a polytope,
i.e., P cfw_ x :

C&O 355
Mathematical Programming
Fall 2010
Lecture 5
N. Harvey
Review of our Theorems
Primal LP:
Dual LP:
Fundamental Theorem of LP: Every LP is either
Infeasible, Unbounded, or has an Optimal Solution.
Not Yet Proven!
Weak Duality Theorem: If x feasib

C&O 355
Mathematical Programming
Fall 2010
Lecture 4
N. Harvey
Outline
Solvability of Linear Equalities & Inequalities
Farkas Lemma
Fourier-Motzkin Elimination
Proof of Farkas Lemma
Proof of Strong LP Duality
Strong Duality
Primal LP:
Dual LP:
Strong Dual

C&O 355
Mathematical Programming
Fall 2010
Lecture 3
N. Harvey
Image: http:/www.flickr.com/photos/singapore2010/4902039196/
C&O 355
Mathematical Programming
Fall 2010
Lecture 1
N. Harvey
Duality: Geometric View
Suppose c=[-1,1]
Then every feasible x sat

C&O 355
Mathematical Programming
Fall 2010
Lecture 2
N. Harvey
Outline
LP definition & some equivalent forms
Example in 2D
Theorem: LPs have 3 possible outcomes
Examples
Linear regression, bipartite matching, indep set
Solutions at corner points
Du

C&O 355
Mathematical Programming
Fall 2010
Lecture 1
N. Harvey
What is Optimization?
Basic Idea: From a certain set of objects,
choose the best one.
Example: Getting married
objects = cfw_ all possible spouses
best one this is very hard to make preci

CO 355 Mathematical Optimization (Fall 2010)
Assignment 5
Due: Thursday, Dec 2nd.
Policy. No collaboration is allowed. You may use the course notes / textbook, lecture slides, and any
solutions to previous assignments but please be very specic when using

CO 355 Mathematical Optimization (Fall 2010)
Assignment 4
Due: Thursday, November 11th, in class.
Policy. No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specic when using citing results foun

CO 355 Mathematical Optimization (Fall 2010)
Assignment 3
Due: Tuesday, November 2nd, in class.
Policy. No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specic when using citing results found

CO 355 Mathematical Optimization (Fall 2010)
Assignment 2
Due: Thursday October 14th, in class.
Policy. No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specic when using citing results found

CO 355 Mathematical Optimization (Fall 2010)
Assignment 1
Due: Tuesday September 28th, in class.
Policy. No collaboration is allowed. You may only use the course notes / textbook and the lecture slides.
Every other resource that you might stumble upon mus

CO 355 Mathematical Optimization (Fall 2010)
Assignment 0 (Preliminary Quiz)
Due: Tuesday September 21st, in class.
Policy. No collaboration is allowed. You are welcome to seek help from the current instructor and TAs
for CO 355.
Question 1: Why did you d

C&O 355
Mathematical Programming
Fall 2010
Lecture 21
N. Harvey
Topics
Max Weight Spanning Tree Problem
Spanning Tree Polytope
Separation Oracle using Min s-t Cuts
Warning!
The point of this lecture is to do things
in an unnecessarily complicated way.

CO 355 Mathematical Optimization (Fall 2010)
Assignment 3
Due: Tuesday, November 2nd, in class.
Policy. No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specic when using citing results found

CO 355 Mathematical Optimization (Fall 2010)
Assignment 4
Due: Thursday, November 11th, in class.
Policy. No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specic when using citing results foun

CO 355 Mathematical Optimization (Fall 2010)
Assignment 5
Due: Thursday, Dec 2nd.
Policy. No collaboration is allowed. You may use the course notes / textbook, lecture slides, and any
solutions to previous assignments but please be very specic when using