OR-MA-ST 706 HW #1 Due Thurs. Jan. 21
Spring 2016
1.
Reiland
Let f:E8 Ä E" be a differentiable function. Show that the gradient vector ff(x) is the vector of partial
derivatives  ``f(x)
x" ,
` f(x)
` x# ,
á,
T
` f(x)
` x8  .
[Hint: In the definition of differentiability, choose B œ B  23 where
2 − I has value 2 − I for component 3 and value ! for all other components. Also recall the definition
of partial derivative:
`0
the partial deriative of 0 ÐB"ß ß B# ß ÞÞÞÞ ß B8 Ñ with respect to B3 , written `B
ß is
3
3
8
"
`0
0 ÐB" ß ÞÞÞ ß B3  ˜B3 ß ÞÞÞ ß B8 Ñ  0 ÐB" ß B# ß ÞÞÞß B8 Ñ
œ 637
Þ]
`B3 ˜B3 Ä !
˜B3
2.
Obtain expressions for ff(x) and f# f(x) for the following functions f:E8 Ä E" :
(i) f(x) œ aT x, where a − E8 is a given constant vector;
(ii) "# xT Ax, where E is an 8 ‚ 8 matrix
(iii) "# xT Ax  bT x, where E is an 8 ‚ 8 symmetric matrix; b − E8 .
3.
Obtain expressions for the gradient ff(x) and Hessian f# f(x) of the function
f(x) œ 100(x#  x#" )#  (1  x" )# .
(This function is called Rosenbrock's function (or the “banana" function because of the shape of its
contours); we will discuss it periodically throughout the course).
4.
Consider the following unconstrained problem:
738373D/ 0 ÐB" ß B# Ñ œ B#"  B" B#  #B##  #B"  /B" B#
a. Write the first-order necessary optimality conditions.
b. Is B œ Ð!ß !ÑT an optimal solution? If not, identify a direction d %I # along which the function would
decrease.
c. Minimize the function 0 starting from Ð!ß !ÑX along the direction d obtained in part b.
5.
Consider the problem to minimize 0 À I 8 Ä I " where 0 ÐBÑ œ EB  ,  œ ÐEB  ,ÑX ÐEB  ,Ñ where E
is an 7 ‚ 8 matrix and , is an 7 ‚ " vector.
a. Give a geometric interpretation of the problem.
b. Write a necessary condition for optimality.
c. Is the optimal solution unique? Why or why not?
d. Can you give a closed form solution of the optimal solution point B.
e. Solve the problem for E and , given below:
#
"
!
E œ
!
"
"
#
"
!
!
"

!
"
#
"
,œ 
"
!
Use the following theorem to do exercise #6:
If f:E8 Ä E" is continuous on a closed and bounded region, then f has a global maximum and
global minimum on that region.
6.
Find the global minimum and global maximum of 0 ÐB" ß B# Ñ œ B#"  B##  $ over  " Ÿ B" Ÿ "ß
 " Ÿ B# Ÿ "
OR-MA-ST 706 HW #1
page 2
Use the following definition and theorem to do exercises #7 and #8 (you don't have to prove the
theorem):
A continuous function 0 ÐBÑ defined on all of E8 is called coercive if
Def.
637
0 ÐBÑ œ  ∞
B  Ä ∞
Thm.
7.
Let 0 ÐBÑ be a continuous function defined on I 8 . If 0 ÐBÑ is coercive, then 0 ÐBÑ has at least one global
minimizing point. If, in addition, the first partial derivatives of 0 ÐBÑ exist on all of I 8 , then these
global minimizers can be found among the critical points of 0 ÐBÑ.
Let 0 ÐB" ß B# Ñ œ B#"  #B" B#  B## Þ
i) show that for each fixed B# , we have
637
0 ÐB" ß B# Ñ œ ∞;
lB" l Ä ∞
ii) show that for each fixed B" , we have 637
0 ÐB" ß B# Ñ œ ∞
lB# l Ä ∞
iii) but 0 ÐB" ß B# Ñ is not coercive.
8.
Let 0 ÐB" ß B# Ñ œ B%"  %B" B#  B%# .
" B#
i) Show that 0 is coercive [0 can be expressed as 0 ÐB" ß B# Ñ œ ÐB%"  B%# Ñ"  B%B% B
% ];
"
#
#
# 
#
ii) show that f 0 ÐB" ß B# Ñ is not positive definite on I evaluate f 0 ÐB" ß B# Ñ at Ð "# ß "# ÑX ;
iii) use the above theorem to show that Ð  "ß  "ÑX and Ð"ß "ÑX are both global minimizers of 0 ÐB" ß B# Ñ.