0. Exercise: old-school. A lumber mill saws both nish-grade and construction-grade boards
from the logs that it receives. Suppose that it takes 2 hr to rough-saw each 1000 board feet of the
nish-grade boards and 5 hr to plane each 1000 board feet of these
Lecture 3
Basic Applications of LP
Dantzig Presents LP George Dantzig developed Linear Programming during World War
II and presented the ideas to a conference of eminent mathematicians and statisticians.
Among the audience were Hotelling and von Neumann.
Lecture 2
Geometry of LPs
Last time we saw that, given a (minimizing) linear program in equational form, one of the
following three possibilities is true:
1. The LP is infeasible.
2. The optimal value of the LP is (i.e., the LP does not have a bounded opt
Lecture 1
LPs: Algebraic View
1.1
Introduction to Linear Programming
Linear programs began to get a lot of attention in 1940s, when people were interested in
minimizing costs of various systems while meeting dierent constraints. We care about
them today b
Lecture 6
Duality of LPs and Applications
Last lecture we introduced duality of linear programs. We saw how to form duals, and proved
both the weak and strong duality theorems. In this lecture we will see a few more theoretical
results and then begin disc
Lecture 5
LP Duality
In Lecture #3 we saw the Max-ow Min-cut Theorem which stated that the maximum ow
from a source to a sink through a graph is always equal to the minimum capacity which
needs to be removed from the edges of the graph to disconnect the s
Lecture 4
Avis-Kaluzny and the Simplex
Method
Last time, we discussed some applications of Linear Programming, such as Max-Flow, Matching, and Vertex-Cover. The broad range of applications to Linear Programming means that
we require ecient algorithms to s
Linear Programming and Semidenite Programming
CMU 15-859E, Fall 2011
Homework 6
Due: Tuesday, Dec 6.
Ground rules: same as for Homework 1. If you have scribed twice, do 2 out of 5 problems. If
you will only scribe once, do 5 out of 5 problems.
1. Youve Go
Linear Programming and Semidenite Programming
CMU 15-859E, Fall 2011
Homework 3
Due: Tuesday, October 11.
Ground rules: same as for Homework 1.
0. (Exercises.) Challenge to a Dual. Write down the dual of the following LP:
maximize
subject to
10x1 2x2 + x3
Linear Programming and Semidenite Programming
CMU 15-859E, Fall 2011
Homework 5
Due: Tuesday, November 15.
Ground rules: same as for Homework 1.
Remarks: We say that two relaxations for the same problem are equivalent if they have the
same optimum value o
Linear Programming and Semidenite Programming
CMU 15-859E, Fall 2011
Homework 4
Due: Thursday, October 20.
Ground rules: same as for Homework 1.
Solve Problems 13, and two out of 47
1. Johnny Minimaximilius. Well now give a proof of the minimax theorem. F
Linear Programming and Semindenite Programming
CMU 15-859E, Fall 2011
Homework 1
Out: Tuesday, September 15
Due: Thursday, September 22
Groundrules
Homeworks will sometimes consist of exercises, easier problems designed to give you practice, and
problems
0. Exercise.
Consider the feasible region K dened by the following constraints:
5x1 + 3x2 5
x1 2x2 4
x1 + 2x2 12
x1 + x2 3
x1 , x2 0
(a) What are the vertices of this feasible region?
(b) What is the maximizer for the cost vector cT = (3, 1)? What about c
Linear Programming and Semidenite Programming
CMU 15-859E, Fall 2011
Homework 2
Due: Thursday, September 29
Ground rules: same as for Homework 1.
1. Unemployment. Consider the assignment problem studied in class; i.e., Maximum-Weight
Perfect Matching in a
Lecture 7
Duality Applications (Part II)
In this lecture, well look at applications of duality to three problems:
1. Finding maximum spanning trees (MST). We know that Kruskals algorithm nds this,
and well see a proof of optimality by constructing an LP f