2.098/6.255/15.093J: Recitation 8
October 31, 2006
1
Network Flow
1.1
Network examples
In order to construct a problem as a network problem, we need to determine what would be considered as
the network ow. If we know the ow, it will be easier to determine

2.098/6.255/15.093J: Midterm review
vanxuan@mit.edu, letallec@mit.edu, mardavij@mit.edu
October 24, 2006
Problem 1 (BT 3.19)
10
4
1
1
2 0 0 0
1 0 0
4 0 1 0
3
0 0 1
For each one of the following statements, nd some parameter values that will make the state

Optimization Methods
MIT 2.098/6.255/15.093
Final exam
Date Given:
December 19th, 2005
P1. [30 pts] Classify the following statements as true or false. All answers must be well-justied,
either through a short explanation, or a counterexample. Unless state

Optimization Methods
MIT 2.098/6.255/15.093
Fall 2006
Midterm exam
Date Given:
Date Due:
October 25th, 2006.
You have two hours (academic time, 110 minutes) to complete the exam.
P1. [30 pts] Classify the following statements as true or false. All answers

Prof. P.A. Parrilo
Fall 2005
Optimization Methods
MIT 2.098/6.255/15.093
Midterm exam
Date Given:
October 20th, 2005
P1. [30 pts] Classify the following as true or false. All answers must be justied. Unless stated
otherwise, all LP problems are in standar

2.098/6.255/15.093J Optimization Methods, Fall 2005
(Brief ) Solutions to Final Exam, Fall 2003
1.
1. False. The problem of minimizing a convex, piecewise linear function over a polyhedron can
be formulated as a LP.
2. True. The dual of the problem is max

2.098/6.255/15.093J: Recitation 5
Xuan Vinh Doan, vanxuan@mit.edu
Nelson Uhan, uhan@mit.edu
October 12, 2005
1
Sensitivity analysis
Suppose we are responsible for production planning for a factory that can produce four types of
products. Each of the four

2.098/6.255/15.093J: Recitation 2
Xuan Vinh Doan, vanxuan@mit.edu, Yann Le Tallec, letallec@mit.edu
September 18, 2006
1
A geometric view of polyhedron in standard form
Let P = cfw_x Rn |Ax = b, x 0, with A Rmn and b Rm , be a polyhedron in standard form.

2.098/6.255/15.093J: Recitation 1
Xuan Vinh Doan vanxuan@mit.edu, Yann Le Tallec letallec@mit.edu
September 12, 2006
1
1.1
Linear Programming
Linearity
The general optimization problem is
min f (x1 , . . . , xn )
(1)
s.t. gi (x1 , . . . , xn ) bi ,
i = 1,

2.098/6.255/15.093J: Recitation 10
vanxuan@mit.edu, letallec@mit.edu, mardavij@mit.edu
November 14, 2006
1
Heuristic Methods
Integer programming problems again are hard to solve. Heuristic methods allow us to obtain good
feasible solutions within a reason

2.098/6.255/15.093J: Recitation 13
vanxuan@mit.edu, letallec@mit.edu, mardavij@mit.edu
December 05, 2006
1
Constrained Optimization
Example 1 (Freund 2004) Consider the following problem:
n
ci
i=1 xi
min
s.t.
n
i=1 ai xi
=b
x0
where ai , ci and b are posi

2.098/6.255/15.093J: Recitation 11
November 21, 2006
1
Convex functions
1.1
Convex function
Let f : Rn R. f is convex if
x, y Rn and [0, 1],
f (x + (1 )y) f (x) + (1 )f (y)
f dierentiable, dom f convex set, and for all x, y dom f , there holds
f (y) f (

2.098/6.255/15.093J: Recitation 12
November 29, 2006
1
Constrained Optimization
1.1
KKT necessary condition
Assume that the feasible region is K = cfw_x Rn |gj (x) 0, hi (x) = 0, x is a local minimum, and
I = cfw_j|gj (x) = 0 is the set of active constrai

2.098/6.255/15.093J: Recitation 7
vanxuan@mit.edu, letallec@mit.edu, mardavij@mit.edu
October 26, 2006
1
Network Formulation
Given a network, a directed graph G = (N, A). For each arc (i, j) A, there will be a cost cij per unit
ow and a capacity uij . For

2.098/6.255/15.093J: Recitation 9
November 8, 2006
1
Integer Programming Duality
Integer programming problems are hard to solve in general. However, there are some sets of constraints
that can be solved more eciently than others . The main idea is to rela

2.098/6.255/15.093J: Recitation 3
September 26, 2006
1
Examples
Example 1.1. All parts below refer to the following standard form tableau.
x1
40
x2
x3
x4
x5
4/3 2/3
0
5/3
0
8
1/3
2/3
1
1/3
0
2
2/3
1/3
0 1/3
1
(a) Write a description of the feasible set.

2.098/6.255/15.093J: Recitation 5
October 12, 2006
1
Sensitivity Analysis
Two optimality conditions for a basis are:
1. Feasiblity: B 1 b 0
2. Optimality: c cB B 1 A 0
Local sensitivity analysis requires us to nd conditions under which these two condition