CS149
Introduction to Combinatorial Optimization
Homework 6
Due: 4:00 pm, Wed, Nov 4th
Problem 1
We want to buy some quantity of meat and bread, in order to satisfy some dietary needs. A kg of meat contains 100g of carbs, 100g of fat and 175g of proteins.
Automatic Recording Problem
1
In this programming assignment, you will use linear programming to obtain an upper bound on the objective for the integer automatic recording problem (ARP). Integer ARP is as follows: maximize user satisfaction with the conte
15-750
Midterm 1
Page 1 of 9
15-750: Practice Midterm 1
Name: Email:
Instructions Fill in the box above with your name, and your email address. Do it, now! This exam is open book: That is, you may use two books(Kozen and CLRS), any handouts from class, an
15-750-s09
Practice Final
Page 1 of 6
15-750-S09: Practice Final
Name: Email:
Instructions Fill in the box above with your name, and your email address. Do it, now! This exam is open book: That is, you may use two books (Kozen and CLRS), any handouts from
15-750
Midterm 2
Page 1 of 9
15-750: Practice Midterm 2
Name: Email:
Instructions Fill in the box above with your name, and your email address. Do it, now! This exam is open book: That is, you may use two books (Kozen and CLRS), any handouts from class, a
15-750
Midterm 2
Page 1 of 9
15-750: Practice Midterm 2
Name: Email:
Instructions Fill in the box above with your name, and your email address. Do it, now! This exam is open book: That is, you may use two books (Kozen and CLRS), any handouts from class, a
Note 5: Duality and Dual Simplex Method
If the primal problem (P) is max Z = n=1 cj xj j n s.t. aij xj bi i = 1, 2, ., m j =1 xj 0 j = 1, 2, ., n, then the dual problem (D) is min W = m bi yi i=1 m s.t. aij yi cj j = 1, 2, ., n i=1 yi 0 i = 1, 2, ., m.
(1
Note 3: LP Duality
If the primal problem (P) in the canonical form is min Z = n=1 cj xj j n i = 1, 2, ., m s.t. j =1 aij xj bi xj 0 j = 1, 2, ., n, then the dual problem (D) in the canonical form is max W = m bi yi i=1 m s.t. aij yi cj j = 1, 2, ., n i=1
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 6: LP Duality, Complementary slackness, Farkas Lemma, and von Neumann min-max principle.
Notes taken by Pixing ZHANG February 17, 2005
Summary: In this lecture, we further discuss the dua
MATHcfw_3/M2510 Optimisation 34
Friday, 30 January, 2009
Lecture 3
Lecturer: Aram W. Harrow Farkas Lemma and strong LP duality
3.1
Using reductions between LPs to extend Farkas Lemma
Yesterday we saw how dierent forms of linear programs could be reduced t
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 6 Solutions
1
Separators in outer planar graphs (20 pts.)
(a) We say that a graph is outer planar if it is 1) planar, and 2) there exists a planar embedding such that all
15-750
HW 6
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 6 Due date: Friday, May 1
Some Reminders:
Read the Policies section on the course web site before you start working on this assignment. Collaboration is per
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 5 Solutions
1
Something about Laplacian (25 pts.)
The Laplacian of a graph encodes a number of interesting properties of the graph. In this problem, we will examine some (
15-750
HW 5
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 5 Due date: Wednesday, April 15
Some Reminders:
Read the Policies section on the course web site before you start working on this assignment. Collaboration
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 4 Solutions
1
Minimum Spanning Trees (25 pts.)
While we did not have time to cover them in class, minimum spanning trees are a crucial building block for many more complic
15-750
HW 4
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 4 Due date: Monday, March 30
Some Reminders:
Read the Policies section on the course web site before you start working on this assignment. Collaboration is
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 3 Solutions
1
Duality Theory (25 pts.)
Given a linear program mincfw_c x : Ax b, x 0 and a solution x0 , how can one decide whether x0 is an optimal solution? More general
15-750
HW 3
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 3 Due date: Wednesday, March 18
Some Reminders:
Read the Policies section on the course web site before you start working on this assignment. Collaboration
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 2 Solutions
1
Traveling Salesman (25 pts.)
In the traveling salesman problem we are given n cities and all distances between the cities. A traveling salesman needs to visi
15-750
HW 2
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 2 Due date: Friday, February 20
Some Reminders:
Read the Policies section on the course web site before you start working on this assignment. Collaboration
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 1 Solutions
1
Fibonaccimal Numbers (25 pts.)
Suppose instead of using powers of two, we now represent integers as the sum of Fibonacci numbers. That is, rather than repres
15-750
HW 1
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 1 Due date: Friday, February 6
Some Reminders:
Read the Policies section on the course web site before you start working on this assignment. Collaboration i
15-750
HW 0 Solutions
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 0 Solutions
1
Asymptotic Notation
[10 points] For each list of functions, order them according to increasing asymptotic growth. Provide a brief arg
15-750
HW 0
1
15-750 Graduate Algorithms Spring 2009
Miller and Dinitz and Tangwongsan Assignment 0 Due date: January 23
Some Reminders: Read the Policies section on the course web site before you start working on this assignment. Collaboration is not per
Farkas Lemma
Rudi Pendavingh
Eindhoven Technical University
Optimization in Rn , lecture 2
Rudi Pendavingh (TUE)
Farkas Lemma
ORN2
1 / 14
Todays Lecture
Theorem (Farkas Lemma, 1894)
Let A be an m n matrix, b Rn . Then either:
1 2
There is an x Rn such tha
Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard
LP Methods.S1 Dual Simplex Algorithm
In the tableau implementation of the primal simplex algorithm, the right-hand-side column is always nonnegative so the basic solution is feasib
CHAPTER VIII
CONVEX SETS
We present some geometric aspects of linear programming. Definition. A subset S of n is said to be convex, if given x and y in S, then the line segment [x,y] connecting x and y lies in S. The line segment [x,y] connecting x and y
15 September 2003
Solving Linear and Integer Programs
Robert E. Bixby
ILOG, Inc. and Rice University
Ed Rothberg
ILOG, Inc.
Outline
Linear Programming: Bob Bixby
Example and introduction to basic LP, including duality Primal and dual simplex algorithms Co