2.098/6.255/15.093 - Recitation 9
Michael Frankovich and Shubham Gupta
November 20, 2009
1
Unconstrained Optimization
1.1
Optimality Conditions
Consider the unconstrained problem: minxRn f (x), where f (x) is twice dierentiable,
the optimality conditions
2.098/6.255/15.093 - Recitation 8
Michael Frankovich and Shubham Gupta
November 13, 2009
1
Dynamic Programming
The number of crimes in 3 areas of a city as a function of the number of police patrol
cars assigned there is indicated in the following table:
15.093 - Recitation 6
Michael Frankovich and Shubham Gupta
October 23, 2009
1
BT Exercise 7.1 (Caterer Problem)
Solution
Construct the n/w as follows: For each day i, create two nodes as follows:
node ci for clean tablecloths, with supply ri
node di fo
15.093 - Recitation 5
Michael Frankovich and Shubham Gupta
October 9, 2009
1
BT Exercise 5.5
Solution
The tableau is:
0
0 c3
0 c5
10
1
-1
0
20
0
2
1
31
0
4
0
a) The necessary and sucient conditions for optimality are c3 0 and c5 0.
b) Continuing the simpl
15.093 - Recitation 4
Michael Frankovich and Shubham Gupta
October 2, 2009
1
BT Exercise 4.5
Consider a linear programming problem in standard form and assume the rows of A are
linearly independent. For each of the following statements, provide either a p
15.093 - Recitation 3
Michael Frankovich and Shubham Gupta
September 25, 2009
1
Simplex Full Tableau method
Example 1. Solve the following problem using the full tableau method:
min
s.t.
x1 x2
x1 + x2 2
x1 + x2 0
x1 , x2 0
Solution. We rst rewrite the pr
15.093 - Recitation 2
Michael Frankovich and Shubham Gupta
September 21, 2009
1
Linear Algebra Review
Read Section 1.5 of BT. Important concepts:
linear independence of vectors
subspace, basis, dimension
the span of a collection of vectors
the rank of
15.093 - Recitation 1
Michael Frankovich and Shubham Gupta
September 11, 2009
1 Convex Functions and Convex Sets
A set A is convex if x, y A, we have
x + (1 )y A.
A function f : R R is convex if x, y R, we have
f (x + (1 )y ) f (x) + (1 )f (y ), [0, 1].
Optimization Methods
MIT 6.255/15.093 Fall 2008
Midterm exam
Date Given:
Date Due:
October 16th, 2008.
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 mus
MIT, 2.098/6.255/15.093J
Optimization Methods
Mid-Term Exam, Fall 2009
Solutions
1. This is a 90 minute exam.
2. It is open book, open notes. No computers allowed.
3. Good luck!
Problem 1. (40 Points)
Classify each one of the following as either True or F
Optimization Methods
MIT 2.098/6.255/15.093
Final exam
Date Given:
December 19th, 2006
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 sta
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 p olyhedron can
b e formulated as a LP.
2. True. The dual of the problem is
2.098/6.255/15.093 Optimization Methods
Practice True/False Questions
December 11, 2009
Part I
For each one of the statements below, state whether it is true or false. Include a 1-3 line
supporting sentence or drawing, enough to convince us that you are n