Unconstrained optimization on the plane: necessary condition
See a very simple geometric discussion of the one-dimensional case. It reveals the Taylor decomposition as the main research tool. Therefore we give the Taylor decomposition in a 2D case. Assuming that the reader has familiarized him/herself with that information, we go directly to the decomposition
(1)
Here is a twice-differentiable function,
is an internal point of the domain
,
is a small vector such that
also belongs to the domain,
is a row vector of first derivatives, and
is the Hessian (a matrix of second-order derivatives). stands for transposition.
When there is no local minimum or maximum?
We have seen how reduction to a 1D case can be used to study a 2D case. A similar trick is applied here. Let us represent the vector as
where
is another vector (to be defined later) and
is a small real parameter. Then
will be close to
. From (1) we get
(2)
We think of as fixed, so the two expressions in square brackets are fixed numbers. Denote
. An important observation is that
When tends to zero,
tends to zero even faster.
Therefore the last term in (2) is smaller than the second, and from (2) we obtain
(3)
The no-extremes case. Suppose the vector of first derivatives is not zero: , which means that
(4) at least one of the numbers is different from zero.
Select . Then (3) implies
(5)
From (4) it follows that . Then (5) shows that
cannot be an extreme point. Indeed, for small positive
we have
and for small negative
we have
. In any neighborhood of
the values of
can be both higher and lower than
Conclusion. In case (4) cannot be a local minimum or maximum. In other words, we should look for local extrema among critical points which satisfy the first order condition
The FOC is necessary for a function to have a local minimum or maximum. All of the above easily generalizes to dimensions higher than 2.
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