ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 2 - Quiz 2:
Feb 5
1.
Determine if the following function is convex, concave, or neither.
1
f (x1 , x2 , x3 ) = 2x21 + x1 x3 + x22 + 2x2 x3 + x23
2
Solution:
The Hessian of f is
4
2 f = 0
1
ECE 490
1.
Problem Set 1
(a)
Quiz: Jan 26
Find the inf and sup of the function
(
)
over the set
cfw_
(b)
Does there exist (i) a minimizer and (ii) a maximizer to this problem?
Solution:
The sup is achieved at
So (
dotted lines
()
()
.
) is the maximizer
ECE 490 Midterm Exam I
Spring 2012
March 8, 7:00 - 8:30 p.m.
_
Name and NetID
DO NOT OPEN
UNTIL EXAM BEGINS
Each question is worth 20 points. You have to clearly explain the reasons for your answers to get credit.
1.
Determine if the following function is
ECE 490 Midterm Exam I Solutions
Solution 1:
The Hessian of is
401
2 = 0 2 2.
121
22
02
+ det
= 4(2 4) + (2) = 8 2 = 10.
4 det
21
12
The determinants of the principal minors are 4, 8, and
The Hessian 2 is not positive semi definite. It is also clear t
Introduction to Optimization (ECE 490)
Final exam, 5.9.2016, 8:00-10:30am
Nine problems; 10 pages. No cell phones or internet usage. Show reasonings; box your
answers.
UIN
Name
1 [4] 2 [5] 3 [4] 4 [3] 5 [4] 5 [4] 5 [4] 5 [4] 5 [4]
Problem 1
Which of the f
202
3.2
3.2.1
CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
Limits and Continuity of Functions of Two
or More Variables.
Elementary Notions of Limits
We wish to extend the notion of limits studied in Calculus I. Recall that when
we write lim f (x) = L, we mea
ECE490: Introduction to Optimization
Ql
Time alloted: 15 mins.
Name:
-~
_-e
_W_
-"<'
c.M
'-'-_
_:.Gc_~_o_i_ _ _ _ _ _ _ _ _
1. (20 points) State the necessary and sufficient optimality conditions for unconstrained problems to have a minimum.
s(Att
vf() =-
/
f7' y ).( ~., ( (:-
()
ECE490: Introduction to Optimization
Q3
Time alloted: 15 mins.
Name:
10"-f
Wad
(),)j
1. (40 points) Consider the function f( x) = xi + 2(1- c)x 1 x 2 + ~ Find the condition number
of the Hessian of f.
J-F
/\-:_
-/:~
v J(:r).: \2_
A Karush-Kuhn-Tucker Example
Its only for very simple problems that we can use the Karush-Kuhn-Tucker conditions
to solve a nonlinear programming problem. Consider the following problem:
maximize f (x, y) = xy
subject to x + y 2 2
x, y 0
Note that the fea
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 3 - No quiz
this week
1.
Suppose
M I 2 f (x) M I x,
where M < is a scalar and I is the identity matrix. Prove that
kf (x) f (y)k M ky xk .
You can use the following facts from linear algebr
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 7
1. a) (LY) Consider the problem
min
2x21 + 2x1 x2 + x22 10x1 10x2
s.t.
x21 + x22 5
3x1 + x2 6.
Find x and that satisfy the first-order KKT conditions.
b) Is x the optimal solution to this
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 6
1. (DB) Use the Lagrange multiplier theorem to solve the following problems:
Pn
2
a) f (x) = kxk , h (x) = i=1 xi 1
Pn
2
b) f (x) = i=1 xi , h (x) = kxk
1
2
c) f (x) = kxk , h (x) = xT Qx
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 5
1.
Consider the unconstrained optimization problem minxRn f (x). Assume that f is twice continuously differentiable, and show that if x is a local min, then f (x ) = 0 and 2 f (x ) 0.
Sol
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set I - Quiz: Feb 5
1.
(a)
Find the inf and sup of the function
f (x1 , x2 ) = x1 + 2x2
over the set
D = x : x21 + x22 1, x1 > 0, x2 > 0 .
(b)
Does there exist (i) a minimum and (ii) a maximu
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 8
1. Let f and g be convex functions and S a convex set. Show that
D = cfw_(y, z, w) : Ax b = y, g (x) z, f (x) w, for some x S
is a convex set. Here A is a matrix and b is a vector.
Soluti
ECE 490 - Introduction to Optimization - Spring 2015- Problem Set 9
1. Prove the following properties of the subgradient, where f and g are convex functions:
(a) Scaling: for a > 0, (a f ) = a f.
(b) Addition: (f + g) = f + g.
(c) Affine combination: if h