CS 435 : Linear Optimization
Lecture 22: Primal-Dual Algorithm for Shortest Paths
Fall 2008 Indian Institute of Technology, Bombay
Lecturer:
Computer Science & Engineering
Sundar Vishwanathan
In this lecture we will design an algorithm for the classical s
CS 435 : Linear Optimization
veturerX
Lecture 10:
Ax b as a convex combination of its extreme points
Fall 2008
Computer Science & Engineering
Sundar Vishwanathan
Indian Institute of Technology, Bombay
sn this letureD we omplete the proof of theorem stting
CS 435 : Linear Optimization
veturerX
Lecture 9: Convex Hull of Extreme Points
Fall 2008
Computer Science & Engineering
Sundar Vishwanathan
Indian Institute of Technology, Bombay
sn this letureD we omplete the proof of the theorem on extreme points mentio
CS 435 : Linear Optimization
Lecturer:
Sundar Vishwanathan
Lecture 8: Extreme points
Fall 2008
Computer Science & Engineering
Definition 1
points. Given points
Indian Institute of Technology, Bombay
p1 ; p2 ; p3 ; : : : ; pn , the convex hull is the small
CS 435 : Linear Optimization
Lecturer:
Lecture 7: Maximising
cT x
Over a Convex Set
Fall 2008
Computer Science & Engineering
Sundar Vishwanathan
Indian Institute of Technology, Bombay
1 Convex Sets
Recall the denitions of a convex set from the previous le
CS 435 : Linear Optimization
Lecture 6: Row Rank. Convex Sets.
Fall 2008 Indian Institute of Technology, Bombay
Lecturer:
Computer Science & Engineering
Sundar Vishwanathan
1 Row Rank of a matrix
The space fx : Ax = 0g is called the null space of the matr
CS 435 : Linear Optimization
Lecture 5: Nullspace. Column Space.
Fall 2008 Indian Institute of Technology, Bombay
Lecturer: Sundar Vishwanathan
Computer Science & Engineering
Consider the subspace fx : Ax = 0g. This is called the nullspace of the matrix A
CS 435 : Linear Optimization
Lecture 4: Linear algebra : Basis, Dimension
Fall 2008 Indian Institute of Technology, Bombay
Lecturer: Sundar Vishwanathan
Computer Science & Engineering
1
Basis (contd.)
We rst give a proof of the result stated in the previo
CS 435 : Linear Optimization
Lecture 3: Linear Algebra Basics
Fall 2008
Lecturer: Sundar Vishwanathan
Computer Science & Engineering
Indian Institute of Technology, Bombay
Last time we saw how to solve a system of linear equations. To understand what the
CS 435 : Linear Optimization
Lecture 2: Linear Algebra: Solving
Ax = b via Gaussian Elimination
Fall 2008
Lecturer: Sundar Vishwanathan
Computer Science & Engineering
Indian Institute of Technology, Bombay
1 Formulation
A linear optimization problem can b
CS 435 : Linear Optimization
Lecture 11: The Simplex Algorithm
Fall 2008
Lecturer: Sundar Vishwanathan
Computer Science & Engineering
Indian Institute of Technology, Bombay
We saw last time that to solve the linear programming problem, it was enough to nd
CS 435 : Linear Optimization
Lecture 12: The Simplex Algorithm: Proof of Correctness
Fall 2008 Indian Institute of Technology, Bombay
Lecturer:
Computer Science & Engineering
Sundar Vishwanathan
In this lecture we will analyse the stopping condition of th
CS 435 : Linear Optimization
veturerX Sundar Vishwanathan
Lecture 13: Introduction to Duality
Fall 2008
Computer Science & Engineering
0
Indian Institute of Technology, Bombay
0 00 00
vet x0 e n extreme pointF uppose it is given y A x0 a b ; A x0 < b F e
CS 435 : Linear Optimization
Lecture 20: Primal-dual algorithm for MST: The Algorithm
Fall 2008 Indian Institute of Technology, Bombay
veturerX Sundar Vishwanathan Computer Science & Engineering
e ontinue with the jo of designing priml dul lgorithm for ws
CS 435 : Linear Optimization
Lecture 20: Primal-dual algorithm for MST
Fall 2008 Indian Institute of Technology, Bombay
veturerX Sundar Vishwanathan Computer Science & Engineering
yur ojetive is to design omintoril lgorithms for spei prolems using hints f
CS 435 : Linear Optimization
Lecturer:
Lecture 19: Using LP techniques to design algorithms for combinatorial problems
Sundar Vishwanathan
Fall 2008
Computer Science & Engineering
Indian Institute of Technology, Bombay
1 Integer Linear Programs
An Integer
CS 435 : Linear Optimization
Lecturer:
Lecture 18: An Algorithm for Matching
Fall 2008
Computer Science & Engineering
Sundar Vishwanathan
Indian Institute of Technology, Bombay
1 Recap of previous lecture
To make the discussion of the previous lecture pre
CS 435 : Linear Optimization
Lecturer:
Sundar Vishwanathan Computer Science & Engineering
Lecture 17: Graph Algorithms: Matchings
Fall 2008
Indian Institute of Technology, Bombay
In this lecture we reviewed old material. Mostly data structures. We noted t
CS 435 : Linear Optimization
Lecturer:
Lecture 16: Duality: Bounding the Primal Solution from Below
Sundar Vishwanathan
Fall 2008
Computer Science & Engineering
Indian Institute of Technology, Bombay
In this lecture we will see another motivation for dual
CS 435 : Linear Optimization
Lecture 15: Complementary Slackness
Fall 2008
Lecturer: Sundar Vishwanathan
Computer Science & Engineering
Indian Institute of Technology, Bombay
1 Complementary Slackness
Primal max cT x Dual min y T b AT y = c y0
Ax b
We pro
CS 435 : Linear Optimization
Lecturer: Sundar Vishwanathan
Lecture 9: Duality Theorem
Fall 2008
Computer Science & Engineering
Indian Institute of Technology, Bombay
Here is the duality theorem.
1 Duality
Theorem 1 If the primal is feasible and has a nite
CS 435 : Linear Optimization
Lecturer: Sundar Vishwanathan
Lecture 1: Introduction
Fall 2008
Computer Science & Engineering
Indian Institute of Technology, Bombay
This subject rst came up for study in the manufacturing and industrial engineering areas in