Re: Possible Calculus of Variations Problem

On May 6, 3:48 am, lugit...@xxxxxxxxx wrote:
On May 5, 7:06 pm, Ray Vickson <RGVick...@xxxxxxx> wrote:



On May 5, 3:50 pm, lugit...@xxxxxxxxx wrote:

Consider some function of two variables f(x,y).  In general, how would
I find a function g(x) such that f(x,g(x)) is as great as possible for
all values of x, subject to the constraint that a<g(x)<b for all x?

There may not be any such function. However, if you use non-strict
inequalities a <= g(x)x <= b, the problem is straightforward (at least
for continuous, bounded f): for each fixed x, let M(x) = value of y in
[a,b] that maximizes f(x,y). Set g(x) = M(x) to make f(x,g(x)) as
large as possible at x. Similarly, if m(x) = value of y in [a,b] that
minimizes f(x,y) and set g(x) = m(x) to minimize f(x,g(x)). Calculus
of variations has nothing to say about this problem.

For f having discontinuities and/or unboundedness, there may not be
any solution.

Similarly, how would I find a function g(x) such that f(x,g(x)) is as
small as possible for all values of x, subject to the constraint that
a<g(x)<b for all x? Please note that in general, f may have
discontinuities: for instance f(x,y)=1/xy has (the equivalent of) a
vertical asymptote at (0,0).

OK, but for each fixed x > 0, 1/xy is maximized at y = a (if a > 0)
and is minimized at y = b.  Things are different if a < 0 < b. In that
case, for x > 0 there is no maximum or minimum of 1/xy. Still, though,
calculus of variations (as usually understood) has nothing to do with
the problem.

R.G. Vickson

Since this problem involves finding a function that satisfies a given
extremizing condition, I am guessing it is a calculus of variations
problem.  However, I'm not entirely sure.

The reason I'm asking is that I'm trying to create an easy way to find
Dini derivatives.

Any help would be greatly appreciated.

Thank you in advance.

Actually, it is indeed the closed interval [a,b] that I'm interested
in.  So how would I find M(x) and m(x), as you call them, given a
function f(x,y) and two real numbers a and b such that a<b?  In other
words, how would I find the values of y which respectively maximize
and minimize the function f(x,y) for a fixed x?  

If, for given x, the function h(y) = f(x,y) is continuously
differentiable on [a.b], the maximum (or minimum) is either at an
interior point, where g'(x) = 0, or at an end point. Necessary
conditions for end-point extrema are: g'(a) <= 0 if a is a (local)
max; g'(b) >= 0 if b is a (local) max; opposite signs to the above if
you want a min. So, the method depends very much on the details. There
are some very difficult 1-dimensional optimization problems, having
thousands of local maxima or minima (where the derivative vanishes)
and possibly also endpoint optima. In the worst case you might need to
find many separate local optima and compare the actual values of g(x)
to find the best one. There are some (numerical) global optimization
methods that work fairly well in one-dimensional cases; my favorite is
Shubert's Method, which works for a Lipschitz function on [a,b] with
known Lipschitz constant; see Bruno Schubert, "A Sequential Method
Seeking the Global Maximum of a Function", SIAM J. on Numerical
Analysis, Vol. 9, No. 3, (Sept. 1972), pp. 479-388. There are numerous
other better methods available, depending on the degree of smoothness
of g(x); a Google search using "Bruno Shubert + optimization" turns up
numerous hits. I have a pdf file of Shubert's paper, and can send it
to you if you tell me your e-mail address.


Do I take the partial
derivative of f with respect to y, set it equal to zero, solve for y,
then among all the values of y that I find,

Basically, yes.

including a and b,

No, not including a and b.

find
the greatest and least values for f(x,y)?

That is it, in a nutshell.


The kind of function I'm interested in is actually of the form f(x,y)=
(h(x+y)-h(x))/y, where h(x) is any function.

I hope you don't really mean /any/ function. If h is smooth on [a,b]
that would help. The more you know about the properties of g, the
better. In the absence of more information you could just use one of
the general methods.

R.G. Vickson

How would I find values
of y which extremize such a function?

.

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