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[EMAIL PROTECTED] (ZHANG Yan) wrote in message news:<[EMAIL PROTECTED]>...
> Suppose that d is positive integer, i.e. d=1,2,3...
>
> A function is defined as follows.
>
> C(d)=C1(d)+C2(d)
>
> Now, I have to proof that there exists an optimal d, see d_{op},
> leading to minimum C(d). And also I have to find an algorithm to find
> d_{op}.
>
>
> I am able to proof that C1(d) is decreasing function of d, and C2(d)
> is increasing function of d. Note that no closed-form expression for
> C1(d) or C2(d), or even has closed-form, its first and second
> derivative is extremely difficult to obtain.
As has been pointed out, since C1 and C2 are defined only on
the positive integers, the derivatives are undefined.
>
> Could you plz give some suggestions to proceed? Many thanks in
> advance.
A minimum over the integers at d_{op} would satisfy a
condition like:
C(d_{op}) <= C(d) for all d.
That's a global optimum. A local optimum would satisfy
C(d_{op}) <= C(d_{op}+1)
C(d_{op}) <= C(d_{op}-1)
Proof of optimality would require verifying these inequalities,
rather than trying to do something with the (undefined)
derivatives.
It has also been pointed out that if the information you gave is
all you know, then this proof is not possible. For instance,
C1(d) = -d^2 is a decreasing function of d>0
C2(d) = d is an increasing function of d.
But C1(d) + C2(d) has no minimum.
So the existence of a minimum clearly does not follow from what you
have stated.
Randy
