\HHT Method of Lagrange Multipliers h\\b\w
\bMethod of Lagrange Multipliers.b\

\bProblem.b\  For given function f(x) where x=(x\[1 \ ,x\[2 \ ,...,x\[n \ ) - n variables,
find an extrema if x belongs to domain H which is set by one or more
constraints:

  g(x)  = 0   or

  g\[1 \ (x) = 0
  g\[2 \ (x) = 0
  ...
  g\[m \ (x) = 0.

\bNote.b\  Summation rule x\[1 \ y\[1 \  + x\[2 \ y\[2 \  + ... = x\[i \ y\[i \  is assumed below.

\bIdea.b\ 
Let's start from case m=1.
As for three dimesional case, 
let's call domain H and contour f(x)=const as "hypersurface".

As in two-dimesional case n=2, suppose that contour of f is "tangent"
to the hypersurface H in point of extrema x=e.
We need to develop more precisely what does this mean "tangent".

In neighbourhood of p, an arbitrary point of hypersurface H,
equation of hypersurface g can be written as:

  Gx=0
  
  if to choose the point p as an origin,
  denote j-th partilal derivative of g with Gj, 
  and shortcut inner product G\[j \ x\[j \  with Gx.

  The last equation means that H is approximated with S, where
  S is vector subspace of IR\]n \ , 
  S is orthogonal to G, and dim S = n-1.
  
    
  Contour's C equation f(x)-const=0 can be written in similar form
  
  Fx=0,
  
  where F = (F\[1 \ , F\[2 \ , ... , F\[n \ ), F\[j \  is partial derivative f by x\[j \ , and
  contour C is approximated with subspace T which is orthogonal to F.

  Now, we can precisely say that T and S are tangent iff
  F and G are parallel, or

  if G is not vanished, there exist number L that   (*)
       F - L*G = 0, or
       grad(f-Lg) vanishes in point p.

  But, why T and S must be tangent?
  At this point, it can be seen that developed language allows to
  formulate condition of extrmality without geometrical language
  which brought us so far.

  Indeed, for p=e, for each x from S, Fx must be zero.
  Otherwise, f(x)-f(0) and f(-x) - f(0) have
  different sign and p is not an extremum.

  This means that F and S are orthogonal which implies (*).
  (As F and G are vectors in n-dimensional space which both
   are orthogonal to (n-1)-dimensional subspace.)

 
 
 



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