Inequality Study 22th file   Update 2010-01-22
index 　 this 　 program 　 DocA 　 Limit 　
XYGraph v2.3 - web page graph   ☜☞   donate   get code
The Cauchy-Schwarz Master Class   J. Michael Steele   ★★★★★
This file is personal home work. No one
proofread. Cannot promise correctness.
If you suspect any view point wrong,
please ask a math expert near by.
Freeman 2009-06-19-10-46
Please use MSIE browser to read this file.
Did not test other browser. This file is
written under MSIE 6.0 




<a name="docA001">　Index begin　Index this file
2009-06-08-19-10 start
This file http://freeman2.com/tute0008.htm
is Freeman's reading notes. Although Freeman
always keep correct view point. But Freeman's
capability is limited, plus no one proofread
this file. Then you can still find wrong view
point. When read, please put question mark as
often as possible. If you suspect any view 
point wrong, please ask a math expert near by.

<a name="docA002">,<a name="textbook">
This file is a note for reading inequality
book written by Professor J. Michael Steele
The Cauchy-Schwarz Master Class  　★★★★★
Below use 'textbook' as abbreviation.
Freeman also read web pages online, and will 
indicate the source URL at discussion point.

<a name="docA003">
Freeman study mechanical engineering. 
Engineering mathematics do not teach 
inequality. Above book is first inequality
book. First time read, it was very hard.
Although high school time learned
  a*a + b*b >= 2*a*b
But this little knowledge do not help.

<a name="docA004">
This file follow textbook chapter section
order, but not continuous, Freeman skip 
those uncertain sections/problems.

This file first function is to learn 
inequality. Second function is to learn
how to use html code to write math 
equations.

<a name="docA005">　Index begin　Index this file
On 2009-01-27-10-08 Freeman accessed
the next page
http://www.sftw.umac.mo/~fstitl/10mmo/inequality.html
save as sftw.umac.mo_text_math_eqn_good.htm
Above page is the main reference for html
math equation.

In order to let reader to build  html math
equation, previous file tute0007.htm page 
end has math symbol and internal code.

This file third function is to display how
to draw curves in web page. The main engine
is XYGraph v2.3 - Technical Figures

Thank you for read Freeman's inequality page.
2009-06-08-19-47 stop

<a name="docA006">
2009-08-23-15-00 start
□　Exercise 1.14 solution

Exercise 1.14 problem (a) is solved by hint
Exercise 1.14 problem (b) is out of LiuHH's
reach. What is "cardinality of a set B⊂Z3" ?
Without clear understand the definition, 
LiuHH is unable to solve problem (b)
LiuHH major in mechanical engineering.
LiuHH is mathematics admirer and outsider.

<a name="docA007">
To solve problems in
The Cauchy-Schwarz Master Class
LiuHH's goal is to solve 50% of the exercise
problems. In reality, solve 40% is good enough.
Please do not be surprise that LiuHH skip some
text section, skip some problems.
2009-08-23-15-12 stop




2009-12-04-17-45
<a name="ch06a001">　Index begin　Index this file
■■Chapter 06: Convexity ~~ The Third Pillar
■　Convex set and convex function

Many natural phenomenon can be described
by equilibrium equation. But they can 
also be described by minimization theory.
Maximization or stationary theory are
equally well tool for different problem.

<a name="ch06a002">
If we have an object function, which is 
to be minimized. In general case, there 
are multiple solutions. Each solution 
is a minimum value point relative to its 
neighborhood. This mean that the solution
is a local solution, may not be global
minimum.

<a name="ch06a003">
For example, assume object function is
 y(x)=sin(x) for x>=0, then 
x=270,630 ... etc (degree) are minimum
value points. Minimum function value is
-1. If you are non-science major reader
please pay attention to next two lines

<a name="ch06a004">
Minimum value points are 270,630,... deg
above is first line, next is second line
Minimum function value is -1

Both line use 'minimum' as adjective.
First line is x=270 degree get minimum
Second line is sin(270)=-1 is smallest
   y(x)=sin(x) value
First  line is input for sin(x)
Second line is output from sin(x)
Both link to minimum, 
one is where function is defined (input),
other one is function value (output).

<a name="ch06a005">
Above example is trivial, all minimum 
are equal -1. In non-trivial case. We 
can still find many local minimum value
answer. Which one is the global minimum?
We have to compare all local minimum and
find the global minimum.
Whether we found all possible local min.
whether there are other min. points we
missed?
These are general minimization process
headache points.

<a name="ch06a006">
Why convex set and convex function are
interesting?
If we have a minimization problem.
If we know this problem has convex set
as its objective function domain (input)
and if this problem has convex function
as its objective function (output) then
<a name="ch06a007">
there are theorems tell us that
convex minimization problem solution is
unique. (therefore global, no second!)
convex minimization is more restricted
than regular minimization. But if we 
can formulate a problem as a convex 
minimization equations, the answer is
worry-free.
2009-12-04-18-35 here

<a name="ch06a008">　Index begin　Index this file
■　What is convex set, convex function?
Objective function's variables defined 
region is called "domain". For example 
function log(x) require x>0, so
log(x) domain is x>0. For a specific 
problem, we can narrow down the domain.
The narrowed domain must stay in x>0.

<a name="ch06a009">
Convex set is a function's domain with
the following requirement:
If D1 is function f(x)'s domain,
if both xa and xb are in D1
if t is a real number in [0,1], then
the point 
  x0=t*xa+(1-t)*xb ---eqn.AP001
is also in domain D1. 
(otherwise, it is not convex set, 
 click ConvexB for 1-D case)

<a name="ch06a010">
Above is one dimension, 
next is two dimension.
If D2 is function f(x,y)'s domain,
if both (xa,ya) and (xb,yb) are in D2
if t is a real number in [0,1], then
the point (x0,y0) defined by next line
  t*(xa,ya)+(1-t)*(xb,yb) ---eqn.AP002
is also in domain D2
<a name="ch06a011">
eqn.AP002 can be written in two x,y
component equation
  x0=t*xa+(1-t)*xb ---eqn.AP003
  y0=t*ya+(1-t)*yb ---eqn.AP004
How about three dimension problem?
Add a z component. For high dimension
it is complicated to write each 
component equation. Many use vector
x for [x,y,z] or [x1,x2,...,xn] for 
higher dimension. Then
eqn.AP002 can be written as eqn.AP001
with the understanding that x is a 
vector represent all variables.
2009-12-04-19-10 here
<a name="convex01"> 2009-12-04-20-30
Star that you can shape
Number of star corners No chord
bottom to top ratio     



<a name="ch06a012">
2009-12-05-10-42 start
Above is a drawing indicate
one dimensional convex set, red straight
line, whole domain only one section.

one dimensional non-convex set, blue
straight line, whole domain has two 
sections. because of two disconnected
section, if pick first in-domain point
from left section and pick second in-
domain point from right section. Then
straight line connect these two points 
must pass undefined section. This 
situation cause non-convex set.

<a name="ch06a013">
First quadrant blue star has similar
situation. From two in-domain points
draw a straight line, not whole line
stay in domain. This situation cause 
non-convex set.

Red circle (two dimensional domain)
and red line (one dimensional domain)
are all convex set.
2009-12-05-10-52 here



<a name="ch06a014">　Index begin　Index this file
2009-12-05-18-22 start
■　What is the difference between 
convex set 
and 
convex function ?
All function use parameter and give us
a result. Function parameter is input
values to function. The result we get
is function output.

<a name="ch06a015">
convex set is defined parameters which
satisfy convex set rules.
convex function is a transformation
formula which satisfy convex function
rules.

When we talk about convex set, we talk
about function input variables.
When we talk about convex function, we 
talk about function output values.

<a name="ch06a016">
The definition of a convex set (input)
use a straight line.
The definition of a convex function
(output) also use a straight line.

Let us see convex set definition from
online professional web page.
(compare with LiuHH wrote eqn.AP001)
2009-12-05-18-38 here

<a name="ch06a017">
2009-12-03-13-04 LiuHH access next page
http://www.stanford.edu/class/cs229/section/cs229-cvxopt.pdf
page 1/14 (168,829 bytes)
[[
2 Convex Sets
We begin our look at convex optimization
 with the notion of a convex set.
<a name="ch06a018">
Definition 2.1 A set C is convex if, for 
any x, y ∈ C and θ ∈ R with 0 ≤ θ ≤ 1,
  θx + (1 − θ)y ∈ C ---eqn.AP005

Intuitively, this means that if we take 
any two elements in C, and draw a line 
segment between these two elements, then
every point on that line segment also 
belongs to C. 
<a name="ch06a019">
Figure 1 shows an example of one convex 
and one non-convex set. The point
  θx+(1−θ)y  ---eqn.AP006
is called a convex combination of the 
points x and y.
]]

Figure 1 is same as ConvexA drawing

<a name="ch06a020">　Index begin　Index this file
■　Straight line? what ill if use curve?
Here is a question:
Why use straight line to define convex
set? If use curve connect two in-domain
points and go around a corner, what is 
the disappointing result?

<a name="ch06a021">
LiuHH found above question, search for
web pages, use Google exact key word
convex "Jensen's inequality" filetype:pdf
Results 1 - 30 of about 32,700 for convex "Jensen's inequality" filetype:pdf with Safesearch on. (0.43 seconds) 
and
if "not a convex" set OR function filetype:pdf
 Results 1 - 30 of about 1,700,000 for if set OR function "not a convex" filetype:pdf with Safesearch on. (0.49 seconds) 
and
if "not a convex" optimiz OR minimiz filetype:pdf
 Results 1 - 28 of 28 for if optimiz OR minimiz "not a convex" filetype:pdf with Safesearch on. (0.46 seconds)
and
if "were not convex" optimization OR minimization filetype:pdf
Results 1 - 23 of 23 for if optimization OR minimization "were not convex" filetype:pdf with Safesearch on. (0.38 seconds)

<a name="ch06a022">
read many pages, no one direct talk
about above question. However, from 
reading, LiuHH understand few points
as following
1. if a problem use convex set (domain)
   and use convex function. Then answer
   is unique. No second answer.
2. if not use straight line to define
   convex set. then unique answer is
   NOT guaranteed.
<a name="ch06a023">
3. if not use straight line to define
   convex function. then minimum answer
   is NOT guaranteed. If we use a hill
   top curve replace straight line to
   define convex function, if function
   is a smaller hill than the hill curve
   we use, We could get a local maximum 
   when search for minimum.

<a name="ch06a024">
Next three drawing buttons "ConvexB",
"ConvexC", "ConvexD" illustrate above
point 1 and 2. Point three is easy to
realize.
2009-12-05-19-33 here

<a name="ch06a025">
"ConvexB" is one dimensional problem.
Function has just one variable x.
Constraint to x is
  x*x>=1 ---eqn.AP007
  x*x<=4 ---eqn.AP008
These two constraint created non-convex
domain [-2,-1] and [1,2]. Two sections
are disconnected. Each section has a 
minimum function value point. Minimum 
point is not unique.

<a name="ch06a026">　Index begin　Index this file
"ConvexC" Draw an objective function
  Obj1= x*x+y*y  ---eqn.AP009
Here Obj1 value is non-constant.
If Obj1 were a constant, eqn.AP009
is an equality constraint equation.
Not an objective function any more!
If require Obj1=5, how can I minimize
a given 5 value? Objective function 
value must be variable.
Each equal-value-objective contour is
a red dot circle. On the other hand
<a name="ch06a027">
constraint equation constant value
can NOT be changed ! We always see
objective function contour curves.
We never see constraint equation 
"contour curves". 
(just one value, contour what?)


<a name="ch06a028">
In the drawing (click ConvexC), the 
thicker red circle tangent with blue 
circle. This is minimum value answer.
Blue circle is function variable x,y
allowed region. It is called domain 
of function Obj1.
<a name="ch06a029">
Blue circle is convex set. Any two 
points in domain connect a straight
line. Whole line stay in domain.
We can find only one minimum value 
point "A". Second minimum point is 
impossible in convex minimization
problem.

<a name="ch06a030">
"ConvexD" same as "ConvexC" but blue
circle change to a new moon shape.
This is non convex set. Straight 
line from point B to point A pass
undefined outside region. 
Point A is a global minimum. 
Point B is a local  minimum. 
Answer uniqueness disappeared.
If shorter arc between A and B change
to a straight line boundary, then 
local minimum B can not hide from
global minimum A. Local minimum B
can not exist, we re-gain answer
uniqueness.

<a name="ch06a031">
"ConvexC" and "ConvexD" are top view 
of a 3-D problem. Objective function
(red circles) look like an ice cream
cone. Blue domain is a cookie put next
to the cone. We want find lowest point
on the ice cream cone direct above the 
cookie. (this is constraint)
<a name="ch06a032">
If function domain has no limit, then
global minimum is 0 at (x,y)=(0,0)
0*0+0*0=0. But with constraint (above
cookie), (x,y)=(0,0) is excluded. Must
find a different answer.
for "ConvexC" min f(x)=2.5 at point "A"
for "ConvexD" min f(x)=3.0 at point "A"
2009-12-05-20-12 stop
<a name="convex02"> 2009-12-05-11-15
Ice cream cone with cookie (click "Draw ConvexC")
ConvexB 　 ConvexC 　 ConvexD 　
    



<a name="ch06a033">　Index begin　Index this file
2009-12-06-09-50 start
■　convex set verify-equation is a
　　straight line
Now explain that convex set equation
  x0=t*xa+(1-t)*xb ---eqn.AP001
is a straight line. Assume we have 
3-D space point A=(xa,ya,za)
and space point B=(xb,yb,zb)
Above two points are given within a
function domain.
<a name="ch06a034">
For space point C=(xc,yc,zc) satisfy
convex set equation eqn.AP001, 
we have
  xc=t*xa+(1-t)*xb ---eqn.AP010
  yc=t*ya+(1-t)*yb ---eqn.AP011
  zc=t*za+(1-t)*zb ---eqn.AP012
Here point A and B are given, both
be constant. 't' is a variable, and 
point C vary following 't'.
<a name="ch06a035">
If (xa,ya,za) and (xb,yb,zb) are 
arbitrary, it is not an evident 
explanation. We can 
put point A at (xa,ya,za)=(0,0,0) and
put point B at (xb,yb,zb)=(1,0,0)
without loss generality. 
<a name="ch06a036">
(or equivalently, to say
we put coordinate system original
point [0,0,0] on point A, and put
coordinate system [1,0,0] on B)
<a name="ch06a037">
eqn.AP010 to eqn.AP012 simplified to
  xc=t*0+(1-t)*1 ---eqn.AP013
  yc=t*0+(1-t)*0 ---eqn.AP014
  zc=t*0+(1-t)*0 ---eqn.AP015
Point C's y component yc is always
zero, see eqn.AP014
and z component zc is always zero
see eqn.AP015.
<a name="ch06a038">
Point C vary on x-axis, eqn.AP013.
Therefore point C must stay on a 
straight line between A and B.
(assume we have a straight x-axis)
2009-12-06-10-18 stop

<a name="ch06a039">　Index begin　Index this file
2009-12-06-12-23 start
■　Straight line help proof, go 
　　around the corner curve not help

Another reason that convex set 
definition must use straight line 
and can not use curve between A 
and B, can not walk around the 
corner. The reason is that to 
prove convex problem (convex set
for domain and convex function for
output) has unique solution, we 
<a name="ch06a040">
need draw a straight line from 
point B to point A see ConvexD
drawing. Only straight line help
us finish the proof. Curve walk 
around the corner can not prove
the uniqueness theory. Reader
can find this proof in any optimal
design textbook or linear/nonlinear
programming textbook.
2009-12-06-12-34 stop



<a name="ch06a041">
2009-12-06-10-55 start
Above is all about convex set.
A function variable defined space is
called domain. If a domain satisfy
eqn.AP001 or eqn.AP005 condition,
this domain is called convex set.

<a name="ch06a042">　Index begin　Index this file
■　Convex set and convex function has 
　　different dimension
convex set is function's input.
convex function is function's output.
they are quite different. For example
a gravity equation, input is distance
and mass, output is force. Second
example, if we use a salary function,
its input is working time in hour unit
its output is salary in dollar unit.
Input distance output force are different.
Input hour output dollar are different
Main point is that convex set and
convex function are two different
term dimension, in two different world.

<a name="ch06a043">
If a function can be expressed by a 
mathematics equation, we can draw it.
Not all functions are convex. First,
we need to define what is a convex 
function.

<a name="ch06a044">
2009-12-01-21-26 LiuHH accessed
http://www.cse.yorku.ca/~kosta/CompVis_Notes/jensen.pdf
page 1/2 has next ( 91,989 bytes)
[[
Definition Let f(x) be a real valued
 function defined on the interval
 I = [a, b]. f is said to be
convex if for every x1,x2 ∈ [a, b] 
and 0 ≧ λ ≧ 1, (typing error, LiuHH note)
f(λx1 + (1 − λ)x2) ≧ λf(x1) + (1 − λ)f(x2)
<a name="ch06a045">
LiuHH change from above to next
and 0 ≦ λ ≦ 1,
   f(λx1 + (1 − λ)x2) ≦
  λf(x1) + (1 − λ)f(x2) ---eqn.AP016

A function is said to be strictly convex
 if the inequality is strict for x1≠x2.
(LiuHH inserted "in" at above line)

<a name="ch06a046">
Definition f(x) is said to be concave
 (strictly concave) if −f(x) is convex
 (strictly convex).
Intuitively, the definition of convexity
 states that function falls below never
 above the straight line between the 
points (x1, f(x1)) to (x2, f(x2))
 (see Fig. 1).

<a name="ch06a047">
Theorem 0.1 If f''(x) exists on [a, b]
 and f''(x) ≧ 0 on [a, b] then f(x) is
 convex on [a, b].
]]
2009-12-06-11-39 here

<a name="ch06a048">　Index begin　Index this file
2009-12-06-12-03 start
Above section is copied from
http://www.cse.yorku.ca/~kosta/CompVis_Notes/jensen.pdf
LiuHH modified few points, reader should
judge whether these modifications are
correct.
<a name="ch06a049">
The reason to include yorku.ca/~kosta
page is that this file page 2/2 has a
nice graph 
Figure 1: Illustrative example of convexity.
It point out
f(x2)
λf(x1) + (1 − λ)f(x2)
f(x1)
f(λx1 + (1 − λ)x2)
four value exact locations.
Not seen in other collected pages.

<a name="ch06a050">
Above is yorku.ca/~kosta web page.
Next is ee.ucla.edu/ee236b web page

2009-12-01-21-18 LiuHH accessed
http://www.ee.ucla.edu/ee236b/lectures/functions.pdf
page 1/16 has next (156,541 bytes)
[[
<a name="ch06a051">
Definition
f : Rn → R is convex if dom f is a 
convex set and
  f(θx + (1 − θ)y) ≤
 θf(x) + (1 − θ)f(y) ---eqn.AP017
for all x, y ∈ dom f , 0 ≤ θ ≤ 1

graph here, can not copy

<a name="ch06a052">
• f is concave if −f is convex
• f is strictly convex if dom f is convex and
f(θx + (1 − θ)y) < θf(x) + (1 − θ)f(y)
for x, y ∈ dom f , x 6= y, 0 < θ < 1

Convex functions 3–2 
]](last line is page bottom note)

<a name="ch06a053">　Index begin　Index this file
■　Convex set straight line vs.
　　Convex function straight line
Convex set definition use straight line.
Convex function definition use straight
line too.
The difference is that
<a name="ch06a054">
Convex set definition straight line not 
       involve inequality. Just say all
       points on line belong to domain.
Convex function definition straight line
       DO involve inequality. Require
       that function curve/surface below
       straight line. Below is less than.
       Below is inequality.
(certainly, Convex set straight line is
 where you spend time work for living,
 Convex function straight line is where
 you receive your salary.)
2009-12-06-12-21 stop
<a name="convex03"> 2009-12-06-16-40
■　Convex set and convex function
<a name="Jensen01"> 2009-12-10-20-52 　 　Index begin　Index this file
■　Jensen's Inequality Program How to use
Output may contain error, Please verify first
Program environment is MSIE 6.0, please use MSIE

<a name="randInp"> 123 , 321 , 213
Random range less than one, less than ten,
or 10^ ; random number +/0 , +/0/-
random number has digits ; integer only
Each seq. has numbers  
If sequence 1 proportional to sequence 2, inequality become equality.
proportional = (box12 change)
Box11 for 0＜m＝x1≦x2≦...≦xn＝M＜∞
Box12 for p1+p2+...+pn＝1 where pk≧0 for all k=1,2,...,n
Box11, input
Box 12, input
  control center


x min: , x max: ; y min: , y max: ;
x min, x max, y min, y max is coordinate axis range
Graph title:
domain left: , domain right: ; steps:
domain left and domain right is where function defined.
domain left <= x left <= x right <= domain right
If steps=60, draw one curve by 60 short lines. Fast vary function
need larger steps. For example gamma(x) in x=[-5,0]
x left, x right are chord two ends   W: H:
x left: , x right: , t value: t∈[0,1]
f(x) =
f'(x) =
f''(x)=
  Show label
Example: convex,
concave, ; both, 　



You can not draw other curve here. But you can
goto [Modify 606] define your equation
and click [Draw] within yellow stripe.
(Do not click [Draw 606]) 2009-10-08-17-08
<a name="box23">
Box 23,

Box 24, Top 5 lines give Jensen inequality answer


<a name="JSMathList">
●●Javascript math function 

<a name="ch06a055">　Index begin　Index this file
2009-12-07-17-26 start
■　Two interpolations, one input, one output
Let us look at 
  f(θx + (1 − θ)y) ≤
 θf(x) + (1 − θ)f(y) ---eqn.AP017
for all x, y ∈ dom f , 0 ≤ θ ≤ 1

AP017 and AP018 are same equation,
AP018 added sub 0 to distinguish x0
from general variable x and mid 
point xe. AP018 is used in graph 
ConvexE example 4.

<a name="ch06a056">
 　f(θx0 + (1 − θ)y0) ≤
 θf(x0) + (1 − θ)f(y0) ---eqn.AP018
 
This equation is a linear interpolation
for input x as function of θ
  θ*x0+(1−θ)*y0 ---eqn.AP019
x0 and y0 are given point on x-axis
x0, y0 are constant, can not be changed.

A linear interpolation for output
f(x) as function of θ
[ yes ! f(x) become f(θ) , x is locked
  and  θ is free to vary. f() always
  serve for variables ! 200912081208]
f(θ)=θ*f(x0)+(1−θ)*f(y0) ---eqn.AP020
xe = θ * x0 +(1−θ)*  y0  ---eqn.AP019
put them side by side for easy compare.

<a name="ch06a057">
eqn.AP019 generate a mid point xe, 
which is in function domain (x-axis, 
input, working hour.)
xe will be used to evaluate function
value
  f(xe)=f(θx0+(1−θ)y0) ---eqn.AP021
[ f(xe) is on function axis, salary]
Since f(x) is a bowl shape curve.
f(xe) is on the curve
(point 'C' ConvexE example 4)
function curve is below straight 
line between points A and B.

<a name="ch06a058">
Above find point 'C' on curve.
Below find point 'D' on straight
      line AB
The expression
 θf(x0) + (1 − θ)f(y0) ---eqn.AP022
say 
at x=x0 get f(x)=f(x0) vertical pole
at x=y0 get f(x)=f(y0) vertical pole.
<a name="ch06a059">
Both pole are not moving (constant)
On two pole tops, connect a straight
line AB, and θ is a variable. When θ
change we have point move along AB.
Both AP021 and AP022 use same θ.
Both AP021 and AP022 are vertical
on/below each other.

<a name="ch06a060">
For a convex problem, we need convex set.
Straight line value eqn.AP022 is higher.
Curve function value eqn.AP021 is lower.
Bowl shaped f(x) is Jensen inequality's
basic reason.
2009-12-07-18-17 stop

<a name="ch06a061">　Index begin　Index this file
2009-12-08-12-10 start
For a concave problem, we need convex set
and a concave function, that is
function curve go above of straight line.
Jensen inequality reverse direction.

If function vary like sine wave for many
period, then that is 'no problem' !
Jensen go home take a nap !

<a name="ch06a062">
■　Convexity summary
A problem is convex or concave or neither
it HIGHLY depends upon function domain
definition. 
If function domain is a convex set AND
If f''(x) ≧ 0 for all x in function domain,
it is convex problem. OR
If f''(x) ≦ 0 for all x in function domain,
it is concave problem. OR
If f''(x) ≦and≧ 0 for x in function domain,
it is not convex and not concave problem.
2009-12-08-12-30 stop

2009-12-08-12-30 done proofread
2009-12-08-17-41 done spelling check

2009-12-10-16-41 start
<a name="ch06a063">
■　How to use Jensen's Inequality program
Suppose that f:[a,b]→real is a convex
function and suppose that the non-
negative real numbers pj, j=1,2,...,n
satisfy
  p1+p2+...+pn＝1 ---eqn.AQ010
show that for all xj in [a,b], 
j=1,2,...,n one has
<a name="ch06a064">
 
f (
j=n
∑
j=1
 pjxj ) ≦
j=n
∑
j=1
 pjf(xj)
---page 87 line 19
---eqn.6.2
width of above equation 
<a name="ch06a065">
eqn.6.2 is Jensen's Inequality. 
On 2009-12-10 LiuHH rewrite from
Convex set and convex function 
to
Jensen's Inequality Program 
Add function to work with eqn.6.2.
To test Jensen's Inequality Program 
work as following

<a name="ch06a066">
Please click "random#2" (below box11), 
fill in 'x' array and 'p' array. Fill 
convex/concave equation to [f(x) =] box,
or click example 0 to 6. Click "Draw
 JensenA". Box 23 has debug, Box 24 has
Jensen output. Example '6' use gatef() 
to combine two equations to one. 
<a name="ch06a067">
An error occur (convex/concave reverse)
if use gatef(x, 4,8) and Box11 has x>8
Change gatef(x,4,8) to gatef(x,4,80) 
or reduce x array to x<8, get correct 
answer. If suspect convex/concave
reversed, use hand calculation to check,
that is best method. 
If check [Show label] box you may need 
to enlarge the x-axis/y-axis min/max 
range. 9812111150

<a name="ch06a068">
If click [random#2] button, 
Box11 get x-array, x points in domain.
Box12 get p-array, probability sum to 1
Then click [Example 0] button, program
draw f(x)=x*log(x). Now Box11, Box12 and
box [f(x)=] all filled test data. Click
[Draw JensenA] button. The drawing not
change, but Box 24 has output. Top five
lines give Jensen inequality answer.
<a name="ch06a069">
One test run get the next answer
[[
func0=x*log(x)
sumpx=6.2926119560716645
fsumx=11.574480918625841
sumpf=11.775958562069037
*** Convex: fsumx<sumpf
<a name="ch06a070">
iter, xarray, f(x); parray
0, 3.21, 3.7437297082255774; 0.1070806422626364
1, 5.36, 8.99924690644333; 0.3979098803576716
2, 7.65, 15.5654982059641; 0.3129361319736385
3, 7.8, 16.02216512282526; 0.1541786036070683
4, 7.87, 16.236266951318683; 0.027894741798985285
]]

<a name="ch06a071">
func0=x*log(x) is the function in test.
sumpx=6.2926 is p1x1+p2x2+...+pnxn
fsumx=11.574 is sumpx*log(sumpx)
sumpf=11.775 is p1f(x1)+p2f(x2)+...+pnf(xn)
Comparison take place between two 
function evaluations fsumx and sumpf.
*** Convex: fsumx<sumpf
***Concave: fsumx>=sumpf
fsumx is eqn.6.2 left side value.
sumpf is eqn.6.2 right side value.

<a name="ch06a072">
When work with log function, x array
must be all positive number. If create 
x array by random number, make sure 
check [+/0] instead of [+/0/-].
Check box [123 , 321 , 213 ] let you
decide the random number arrangement.
[123 ] output sorted increase sequence.
[321 ] output sorted decrease sequence.
[213 ] output non-sorted sequence.

<a name="ch06a073">
[random#1] button make two sets random
numbers, store in box11 and box12. You
can use this output in other page. 
[random#1] NO probability output.

<a name="ch06a074">
[random#2] button make two sets random
numbers, store in box11 and box12. 
box11 is x array.
box12 is probability array.

[probab#3] button make one set proba-
bility output to box12.

<a name="ch06a075">
Although Liu,Hsinhan try to make every
output correct. But no proofread/use
and no test run for every possible case.
For those forgotten cases, output is
unpredictable. 

2009-12-10-17-36 stop




<a name="docB001">
2009-12-07-18-22 start
From 2009-12-01 to 2009-12-04, 
Liu,Hsinhan goto online find convex set
and convex function papers. Collected 28
pdf files. The following 8 files has a 
short cut for them.

<a name="docB002">
2009-12-01-19-51
http://www.eecs.berkeley.edu/~wkahan/MathH110/Jensen.pdf

2009-12-01-20-05
http://www.recreatiimatematice.ro/arhiva/corespondente/RM22007STEPHAN.pdf

2009-12-01-20-32
http://www.math.ust.hk/excalibur/v5_n4.pdf

<a name="docB003">
2009-12-01-20-46
http://www.nzamt.org.nz/nzimo/wp-content/uploads/2009/01/convex-functions.pdf

2009-12-01-21-22
http://www.soe.ucsc.edu/classes/cmps290c/Spring08/solutions/hw2sol.pdf

2009-12-01-21-29
http://www.maths.bris.ac.uk/~maxmr/opt/convex.pdf

<a name="docB004">
2009-12-03-09-53
http://phildybvig.com/teaching/finopt/problems/finoptp4ans.pdf

2009-12-03-13-04
http://www.stanford.edu/class/cs229/section/cs229-cvxopt.pdf

<a name="docB005">
Recent read text books has

Applied Optimal Design, 
ISBN 0-471-04170-x
Edward J. Haug & Jasbir S. Arora

Introduction to Optimal Design
ISBN 0-07-002460-x
Jasbir S. Arora

<a name="docB006">
Nonlinear Programming
ISBN 0-471-78610-1
Mokhtar S. Bazaraa & C.M. Shetty

Introduction to Linear and
Nonlinear Programming
ISBN 0-201-04347-5
David G Luenberger

<a name="docB007">
These are all textbook LiuHH bought
during University of Iowa studying
period. 1980 to 1990

LiuHH hope this page view points are
all right, but same as always,

2009-12-07-18-50 stop

<a name=20091217>
2009-12-17-11-09 start
Update 2009-12-17 change all tute*.htm
(from tute0007.htm to tute0023.htm)
first:
Correct 'Limit' link from '#docA06'
to '#docA006'
second:
Change Javascript index to read from
jslist1e.js so that update jslist1e.js
then update ALL tute*.htm.
2009-12-17-11-23 stop

<a name=20091224>
2009-12-24-21-54 start
Update 2009-12-24 add example 7, 8
for Problem 6.6 (AMM 2002)
2009-12-24-21-56 stop

<a name=20100122>
2010-01-22-17-57 start
After 'Update 2009-12-24', tute0022.htm
has following change
2009-12-26 update file, added example [9]
2010-01-04 update file, added example [10]
           to [14]
Both update NOT change file top update
sign. Both update are still marked with
'Update 2009-12-24'
Now
'Update 2010-01-22' added example button
[15], [16], [17]
2010-01-22-18-06 stop




<a name="Copyright">　Index begin　Index this file
2009-06-19-10-48
If you are interested in inequality,
suggest you buy the book
The Cauchy-Schwarz Master Class
written by Prof. J. Michael Steele
The Cauchy-Schwarz Master Class is
this web page's textbook.

To buy textbook, that is to show thanks
to Prof. J.M. Steele's great work.
and it is also respect copyright law.
Thank you. Freeman 2009-06-19

The Cauchy-Schwarz Master Class
J. Michael Steele ★★★★★
ISBN 978-0-521-54677-5
2009-06-19-10-56

Javascript index
http://freeman2.com/jsindex2.htm   local
Save graph code to same folder as htm files.
http://freeman2.com/jsgraph2.js   local


This page, Inequality file sixteen.
http://freeman2.com/tute0022.htm
First Upload 2009-12-08

Thank you for visiting Freeman's page.　
Freeman 　2009-12-08-17-51

≦ ≠ ≧ ＜＝＞±≡≈≌≒∏∑√∛∜∝∞⊕⊙
〈v,w〉 ∈ ∀∂⊥∃∋∆∇∟∠∫∬∭∮∥○●◎　
∧∨∩∪∴∵∶∷⊂⊃⊄⊅⊆⊇⊿＋－＊／
§‰¼½¾ ⅓⅔⅕⅖⅗⅘⅙⅚⅛⅜⅝⅞⅟←↑→↓↔↕↖↗↘↙
■□　▢▣▤▥▦▧▨▩▪▫
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡ΢ΣΤΥΦΧΨΩ
ΪΫάέήίΰ　αβγδεζηθικλμνξοπρςστυφχψω