Formula:
When using the simplex method for solving linear programming problems, we have
xB
f
xN
1
1
1
AB AN
AB b
p t p t A 1 AN p t A 1 b
N
BB
BB
Given a vector v and a matrix A, we have
projnull(A) v = I AT (AAT )1 A v
(PAS algorithm) Given the k -th
Math 4553, Homework 6, Due on 4/28/2011
1. (20 points) Consider the smallest circle problem: given n points pi (xi , yi ) on a 2D plane,
nd the smallest circle that contains all these points. For example, in the following graph,
ve points (blue circles) a
Math 4553, Homework 1, Due on 1/28/2011
1. (4 points) A company can produce three dierent types of concrete blocks, identied as A,
B, and C. The production process is constrained by facilities available for mixing, vibration,
and inspection/drying. Using
Math 4553, Solution to Homework 5
1. By typing the following command in Matlab or Octave
p = [-1; -2; 0; 0; 0];
A = [-2,1,1,0,0; -1,2,0,1,0; 1,0,0,0,1];
b = [2;7;3];
Id = eye(5);
beta = 0.9999;
x0 = [0.5; 0.5; 2.5; 6.5; 2.5];
T0 = diag(x0)
y0 = [1;1;1;1;1
Math 4553, Solution to Homework 6
1. Clearly
P=
12342
00201
and the optimization problem can be written as
min
f (c) = cT P T P c (c1 + 4c2 + 13c3 + 16c4 + 5c5 )
n
subject to
ci = 1
i=1
ci 0, for i = 1, . . . , n
Or in the standard form, it is
min
subject
Math 4553, Exam II, Apr. 18, 2011
Name:
Formula:
If x is a local solution to the optimization problem min f (x), subject to x S, where S is a
convex set, then x satises x S and f ( ) (x x) 0 for all x S.
x
(KKT conditions for quadratic programming) If x
Math 4553, Solution to Homework 4
1. To check whether an optimization problem is convex or not, we only need to check to two
things: (1) Is the feasible region convex? (2) Is the objective function convex?
(a) Clearly, the feasible region for this problem
Math 4553, Solution to Homework 3
1. First, we need to form the linear programming problem. Denote eij to be the edge from node
i to node j . Let xij be the ow on eij . If xij = 1, then the route goes through edge eij . If
xij = 0, then the route does not
Solution to Exam II
1. Notice that
Q=
21
,
14
p=
1
,
1
A=
1 2
,
1 1
b=
2
1
(a) Compute the eigenvalues of Q, we have 1 1.5858 and 2 4.4142. Therefore, the
objective function is convex. Hence the quadratic programming problem is convex.
(b) The initial tab
Math 4553, Homework 4, Due on 4/4/2011
1. (10 points) Determine if the following problems are convex optimization problems or not.
(a) Minimize f y) = x2 + y 2 xy 24x 20y
(x,
x + 2y 0
x + 2y 9
Subject to
x+y 8
x+y 0
(b) Maximize f y) = (x 2)2 + (y 10)2
(
Math 4553, Homework 3, Due on 3/23/2011
1. (10 points) Use the simplex method to nd the shortest route from vertex 1 to vertex 9, where
the digraph and distance between cities are given as following:
17
2
10
18
15
1
7
16
10
12
3
5
15
6
16
9
14
13
18
23
16
Math 4553, Exam I, Feb. 25, 2011
Name:
1. (25 points) Consider the following portfolio management problem
Stock type Medicine Precious metal Computer Hardware New Energy Retail
Interest Rate
12%
9%
5%
8%
4%
Risk factor
3.2
1.8
1.6
2.1
1.4
A business man p
Math 4553, Solution to Homework 3
1. First, we need to form the linear programming problem. Denote eij to be the edge from node
i to node j . Let xij be the ow on eij . If xij = 1, then the route goes through edge eij . If
xij = 0, then the route does not
Math 4553, Solution to Homework 4
1. To check whether an optimization problem is convex or not, we only need to check to two
things: (1) Is the feasible region convex? (2) Is the objective function convex?
(a) Clearly, the feasible region for this problem
Math 4553, Solution to Homework 5
1. By typing the following command in Matlab or Octave
p = [-1; -2; 0; 0; 0];
A = [-2,1,1,0,0; -1,2,0,1,0; 1,0,0,0,1];
b = [2;7;3];
Id = eye(5);
beta = 0.9999;
x0 = [0.5; 0.5; 2.5; 6.5; 2.5];
T0 = diag(x0)
y0 = [1;1;1;1;1
Math 4553, Solution to Homework 6
1. Clearly
P=
12342
00201
and the optimization problem can be written as
min
f (c) = cT P T P c (c1 + 4c2 + 13c3 + 16c4 + 5c5 )
n
subject to
ci = 1
i=1
ci 0, for i = 1, . . . , n
Or in the standard form, it is
min
subject
Solution to Exam I
1.
(a) Let xi , i = 1, . . . , 5, be the ammounts of investments on different types of stocks, then
the objective function is
max
f = 0.12x1 + 0.09x2 + 0.05x3 + 0.08x4 + 0.04x5
and the constraints are
x1 + x2 + x3 + x4 + x5 = 1, 000, 00
Solution to Exam II
1. Notice that
Q=
21
,
14
p=
1
,
1
A=
1 2
,
1 1
b=
2
1
(a) Compute the eigenvalues of Q, we have 1 1.5858 and 2 4.4142. Therefore, the
objective function is convex. Hence the quadratic programming problem is convex.
(b) The initial tab
Math 4553, Homework 2, Due on 2/18/2011
1. Consider the problem
min
f = pt x
subject to
Ax b
x0
where
5
9
b = ,
0
3
0 1
1 1
A=
1 2 ,
1 1
1
2
p=
(a) (2 points) Solve the problem using the simplex method.
(b) (3 points) Solve the problem using graphical o
Math 4553, Homework 5, Due on 4/18/2011
1. (10 points) Consider the following linear programming problem
min
subject to
f = x1 2x2
2x1 + x2 + x3 = 2
x1 + 2x2 + x4 = 7
x1 + x5 = 3
xi 0, for i = 1, . . . , 5
Given the starting interior point x0 = (0.5, 0.
Solution to Exam I
1.
(a) Let xi , i = 1, . . . , 5, be the ammounts of investments on different types of stocks, then
the objective function is
max
f = 0.12x1 + 0.09x2 + 0.05x3 + 0.08x4 + 0.04x5
and the constraints are
x1 + x2 + x3 + x4 + x5 = 1, 000, 00