Introduction to Optimization
Grinshpan
Week 6 answers
1.
max min(x y ) = max [] = and min max(x y ) = min [+] = +.
2.
The initial primal dictionary is infeasible. The initial dual dictionary is infeasible as well.
x0
y 0
x 0
w1
w2
w3
y 0
x 0
y 0
=
3y1 +
Introduction to Optimization. HW#5.
Due: Tuesday, December 1, 2014
1. Page 108, Exercise 4-7-1
2. Exercise 4-5-2. (See my solution for Example 4-5-1, posted on the webpage.)
3. With payo matrix A = 1 2 , what is the value of the game? What strategies x an
Introduction to Optimization. HW#4.
Due: Thessday, November 17, 2015
1. Page 95, Exercise 4-2-2.
2. Study the example in Section 4.3. Then solve both the primal and dual LPs simultaneously
using the method in Section 4.2.
3. Page 98, Exercise 4-4-1
4. Pag
Introduction to Optimization. HW#1.
Due: Thursday, Oct 1, 2015
1. (20 points) You are curious about the annual salary of three of your close friends (I just
name them A, B and C.) You somehow got the insider information that the sum of the
three quantitie
Introduction to Optimization.
Final Exam Study Guide
Please pick a seat at random during the nal exam. Please do not sit next to your friend.
1
1. (a)You have available a linear programming software that assumes the input data (A, b, p) species
a linear p
Math 305 Introduction to Optimization
Midterm Examination
1
1. (i) (10 points) Consider the region specified by x1 + x2 > 1 and 3x1 + 2x2 6 6. Use the Phase I
procedure to identify a vertex of this region. Please show all the steps.
(ii) (5 points) Also,
Introduction to Optimization
Grinshpan
Week 1 answers
1. The range of x2 subject to the conditions
x1 + x 2 + x3 = 5
x1 x2 x3
x1 , x 2 , x 3 0
is 0 x2 5/2. Indeed, for every such x2 , the point (0, x2 , 5 x2 ) is feasible.
However, x1 + x2 + x3 > 5 as lon
Introduction to Optimization
Grinshpan
Week 2 answers
1. We have 6x1 + 8x2 + 5x3 + 9x4 = 9 3x1 x2 4x3 9. The maximum value = 9
is attained at the point (0, 0, 0, 1) (and only there).
2. The problem is unbounded. Indeed, the value = t along the feasible ra
Introduction to Optimization
Grinshpan
Week 4 answers
1. Each equation ai1 x1 + . . . + ain xn = 0 describes a hyperplane in Rn . This hyperplane
contains the origin and is normal to
ai1
. .
.
.
ain
Each inequality ai1 x1 + . . . + ain xn 0 describes a h
Introduction to Optimization
Grinshpan
Week 10 answers
y1
1. Suppose that a stochastic vector y = y2 is an optimal strategy for the row player.
y3
Then, since the game is fair, the maximum of the expected payo
E (x) = y Ax
0
1 1
x1
1
0
1 x2
1 1
0
x3
=
Math 305 Introduction to Optimization
1
Makeup Midterm Examination
1. Let S be triangle bounded by the points (1, 1), (1, 2), (2, 2).
(5points)Express S in the form cfw_x = [x1 , x2 ]T R2 : Ax
A and the vector b?
b, x
0, what is the matrix
(5 points) Usin