ELE 704 Optimization
HW7
Due 24 April, 2007
1. Consider the problem
minimize x2 5x + 4
subject to. x2 8x + 12 0
with x R.
(a) Find the feasible set, the optimal value of x and the optimal solution
p of the problem.
(b) Plot the objective and constraint fu
ELE 704 Optimization
HW5
Due 27 March 2007
1. Suppose that x(k) and x(k+1) are two consecutive points generated by the
gradient descent algorithm with exact line search. Show that
T
f x(k)
f x(k+1) = 0
2. For the following quadratic problem
f (x) =
1T
x Q
ELE 704 Optimization
HW4
Due 20 March 2007
1. Prove that
0
mI
H(x)
MI
where H(x) is the Hessian matrix of a strongly convex function f (x) and
m is the smallest and M is the largest eigenvalues of H(x).
2. Write the MATLAB code which performs the followin
ELE 704 Optimization
Homework 2
Due 6 March, 2007
1. Consider the problem of minimizing Ax b
and b Rm
2
where A Rmn , x Rn
(a) Give a geometric interpretation of the problem.
(b) Write a necessary condition for optimality. Is this also a sucient
condition
ELE 704 Optimization
HW3
Due 15 March 2007
1. Prove the following:
(a) If f and g are convex, both nondecreasing (or nonincreasing), and
positive functions on an interval, then f (g (x) is convex.
(b) If f ,g are concave, positive, with one nondecreasing
HACETTEPE UNIVERSITY
DEPT. OF ELECTRICAL AND ELECTRONICS ENGINEERING
ELE 704 Optimization
Midterm Examination, 17 April, 2007
Name
ID #
: Hardworker
: #1
Question
Mark
1
20
2
40
3
4
10
20
5
20
Total
110
Q1. (20pts) A cardboard box for packing some stu is
hcraeseR gnissecorP langiS latigiD rof ertneC
Channel Shortening Filter Design
Based on
Polynomial Methods
Cenk Toker1, Sangarapillai Lambotharan1, Jonathon A. Chambers2, Buyurman Baykal3
1 Centre for Digital Signal Processing Research, Kings College Lond
HACETTEPE UNIVERSITY
DEPT. OF ELECTRICAL AND ELECTRONICS ENGINEERING
ELE 704 Optimization
Midterm Examination, 17 April, 2007
Name
ID #
:
:
Question
Mark
1
20
2
55
3
Total
35
110
Q1. (20pts) Thinking of the Gradient Descent, Steepest Descent and Newtons A
ELE 704 Optimization
HW10
Due 1 June, 2007
1. For the logarithmic barrier method, consider the following problem
min f (x) = log(ex + e
2
s.t. x
1:1x + 0:1
x
)
0
(a) Plot f (x) and t f (x) + (x) for several values of t where x 2 [ 1; 2];
and indicate the
ELE 704 Optimization
HW9
Due 15 May, 2007
1. Newtons method with xed step size = 1 can diverge if the initial point
is not close to x . Consider the following examples:
(a) f (x) = log(ex + ex ) has a unique minimizer at x = 0.
i. Plot f (x) for x [2, 2]
ELE 704 Optimization
HW8
Due 1 May, 2007
1. Show that the weak max-min inequality
sup inf f (w, z) inf sup f (w, z)
zZ wW
wW zZ
always holds, with no assumptions on f : Rn Rm R, W Rn and
Z Rm
2. Consider the following optimization problem
x2 + x2
1
2
min
ELE 704 Optimization
HW6
Due 3 April, 2007
1. Implement Steepest Descent Algorithm in MATLAB with two choices of
quadratic norm for the function given in HW5.
(a) Let P = H(x) where H(x) is the Hessian of the function.
(b) Choose some arbitrary P.
(c) Com