Spring '16
OR-MA-ST 706 HW #4 Due Thurs. April 21
Reiland
1.
If E © I 8 is a subspace of I 8 , show that Ew œ E¼ , where Ew is the dual cone of E and E¼ is the
orthocomplement of E.
Notes:
1. E © I 8 is a subspace of I 8 if for any Bß C − E and any scalars + and , , +B  ,C − EÞ
2. if E © I 8 is a subspace of I 8 , the orthocomplement E¼ ´ ÖB − I 8 À BX C œ ! for all C − E×Þ
2.
If E" and E# are nonempty subsets of I 8 with E" © E# , show that E#w © E"w .
3.
If X œ {ÐB" ß B# Ñ − I # : B#  B$" }, find WÐ\ß BÑ where B œ Ð!ß !ÑÞ
4.
Let \ be a subset of I 8 , and let B − 38> \ . Show that the cone of tangents WÐ\ß BÑ of \ at B is I 8 Þ (A
point B is in the interior of \ , denoted 38> \ , if R% ÐBÑ § \ for some % > 0, where R% ÐBÑ œ ÖC À C  B
 % is an %-neighborhood around the point B).
5.
Consider the minimization problem
Q 38 0 ÐBÑß B − \
where \ © I 8 and 0 À I 8 Ä I " is differentiable.
w
We showed in class that a necessary condition for B to be a minimum of 0 is f0 ÐBÑ − ÐWÐ\ß BÑÑ Þ In the
special case where \ œ I 8 , what is the form of the above necessary condition? (Recall that if O is a cone,
w
then O ´ ÖC − I 8 À CX B ! for all B − O× is the dual cone of O ).
6.
Let 0 À I 8 Ä I " be a convex function. Show that 0 is a subgradient of 0 at B if and only if the hyperplane
{(Bß CÑ À C œ 0 ÐBÑ  0 X ÐB  BÑ× supports /:3 0 at [Bß 0 ÐBÑÓ.
Note:
Let \ be a nonempty subset of I 8 and let B − `\ , where `\ denotes the boundary of \Þ A hyperplane
L œ ÖB À :X ÐB  BÑ œ !× is called a supporting hyperplane of \ at B if either
\ © L  œ ÖB À :X ÐB  BÑ !× or else \ © L  œ ÖB À :X ÐB  BÑ Ÿ !×Þ
(definition: B − `\ if R% ÐBÑ contains at least one point in \ and one point not in \ for every %  !).
Interesting facts relating optimization and 1:
(no work is required for this problem; just read and expand your mathematical horizons).
Everyone knows that 1 œ 3.14159... is the ratio of the circumference of a circle to its diameter, given that
I # is equipped with the usual Euclidean metric. Giving I # other metrics, however, alters the value of 1.
For example, under the taxicab metric ." on the plane, defined by
." ÐÐB" ß C" Ñß ÐB# ß C# ÑÑ œ B#  B"   C#  C" 
circles are diamond-shaped and the value of 1 is 4. For the max-norm metric .∞ given by
.∞ ÐÐB" ß C" Ñß ÐB# ß C# ÑÑ œ 7+BÐB"  B# ß C"  C# Ñ
circles are square and again 1 is 4.
The above metrics are special cases of a class of metrics
.: ÐÐB" ß C" Ñß ÐB# ß C# ÑÑ œ B#  B"   C#  C"  ß
:
:
:
defined only for p 1. The table below gives approximate values of 1 for various values of p.
OR-MA-ST 706 HW #4
:
1:
1
%
1.1
$Þ(&(ÞÞÞ
1.2
$Þ&(#ÞÞÞ
1.5
$Þ#&*ÞÞÞ
1.8
$Þ"&&ÞÞÞ
2
1 œ $Þ"%"ÞÞÞ
2.25
$Þ"&&
page 2
3
$.#&*
6
$.&(#
11
$.(&(
The table suggests, and it is indeed true, that the minimum value of 1: for this class of metrics is
1# œ 1 œ 3.14159...
∞
%