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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "To understand Lagrange mu
ltipliers and how the dual problem arises in maximization or minimizat
ion, let's try to solve a simple problem in one dimension (one the lin
e)." }}{PARA 0 "" 0 "" {TEXT -1 259 "Find one-half of the distance squ
ared from the origin to the point 1.  Ordinarily, one wouldn't use Lag
range multipliers to solve this problem, but we will do so, and easily
 arive at the dual problem.  By analogy, higher dimensional cases will
 become clearer." }}{PARA 0 "" 0 "" {TEXT -1 23 "So we want to minimiz
e " }{XPPEDIT 18 0 "Q(y) = y^2/2;" "6#/-%\"QG6#%\"yG*&F'\"\"#F)!\"\"" 
}{TEXT -1 33 "  subject to the constraint that " }{XPPEDIT 18 0 "y = 1
;" "6#/%\"yG\"\"\"" }{TEXT -1 59 ".  Introduce the multiplier x in the
 Lagrangian as follows:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "L(x,y) = y^2
/2+x*(y-1);" "6#/-%\"LG6$%\"xG%\"yG,&*&F(\"\"#F+!\"\"\"\"\"*&F'F-,&F(F
-F-F,F-F-" }{TEXT -1 265 ".  For different values of x, we obtain diff
erent parabolas in y.  Observe this below in the animation (click on t
he picture and press play to begin,stop to end; by right clicking, you
 can choose options to slow it down, speed it up, and make it play con
tinuously)." }}}{EXCHG {PARA 256 "" 0 "" {GLPLOT2D 496 496 496 
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}}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "To obtain \+
the dual to this problem, we keep track of the lowest point in all the
se different parabolas.  For each parabola obtained, we color the lowe
st point blue.  See this in the animation below.  The green line is al
ways tangent to the paraboal at its lowest point. " }}{PARA 256 "" 0 "
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}{TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "This blue curve represents the \+
dual problem--in this case " }{XPPEDIT 18 0 "-P(x) = -x^2/2-x;" "6#/,$
-%\"PG6#%\"xG!\"\",&*&F(\"\"#F,F)F)F(F)" }{TEXT -1 272 ".  When the bl
ue curve is tangent to the red curve, the problem is solved--that is t
he critical point for the Lagrangian.  Notice that the curves are tang
ent exactly at the highest point of the blue curve.  The maximum of -P
 is the minimum of Q subject to the constraints." }}{PARA 0 "" 0 "" 
{TEXT -1 35 "Also, in this case, the Lagrangian " }{XPPEDIT 18 0 "L(x,
y) = y^2/2+x*(y-1)" "6#/-%\"LG6$%\"xG%\"yG,&*&F(\"\"#F+!\"\"\"\"\"*&F'
F-,&F(F-F-F,F-F-" }{TEXT -1 80 " is a function of two variables and so
 we can see its graph in three dimensions." }}{PARA 256 "" 0 "" 
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10 0 2 4 6 4 2 1.000000 83.000000 -73.000000 0 0 "Curve 1" }}{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "It is a saddle surface and the sa
ddle point is exactly where the the min of Q and max of -P are achieve
d simultaneously." }}{PARA 0 "" 0 "" {TEXT -1 100 "An analogous proble
m in two dimensions would be to find one-half the distance squared fro
m the line " }{XPPEDIT 18 0 "y1 = 1;" "6#/%#y1G\"\"\"" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Lagrangian is " }{XPPEDIT 18 0 "L
(x,y) = (y1^2+y2^2)/2+x*(y1-1);" "6#/-%\"LG6$%\"xG%\"yG,&*&,&*$%#y1G\"
\"#\"\"\"*$%#y2GF.F/F/F.!\"\"F/*&F'F/,&F-F/F/F2F/F/" }{TEXT -1 239 ". \+
 For differenent values of x, we obtain different paraboloids.  If we \+
track the lowest points of these paraboloids, we obtain the dual probl
em. See the animation below.  The green plane is tangent to the lowest
 points of the paraboloids." }{GLPLOT3D 968 555 555 {PLOTDATA 3 "6#-%(
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 545 "In this case, it is extr
emely difficult to visualize the graph of the Lagrangian and notice th
e saddle point, because it is a three-dimensional hypersurface in four
 dimensions.  And in higher dimensions, it even gets harder to see the
 origin of the dual problem.  But we can take our cue from these low-d
imensional cases and generalize the process.  The dual problem arises \+
from taking the hypersurface formed by the lowest points on quadratic \+
hypersurfaces.  Then if we look at the maximum point on this dual hype
rsurface, we solve the problem." }}{PARA 0 "" 0 "" {TEXT -1 154 "These
 examples might foster a false notion that the dual problem is always \+
one-dimensional.  This dimension is exactly equal to the number of mul
tipliers." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }