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In this paper we will generalize the author's two nonzero component lemma to general self-reducing functions and utilize it to find closed from answers for some resource allocation problems.

The technique we will use in this paper was first applied by this author to problems in matrix inequalities and matrix optimization. Historically, many researchers have established matrix inequalities by variational methods. In a variational approach one differentiates the functional involved to arrive at an “Euler equation” and then solves the Euler Equation to obtain the minimizing or maximizing vectors of the functional. The same technique is also often used in matrix optimization. Solving the Euler equations obtained are tedious and generally provide little information. Others have established inequalities for matrices and operators by going through a two-step process which consists of first computing upper bounds for suitable functions on intervals containing the spectrum of the matrix or operator and then applying the standard operational calculus to that matrix. This me- thod, which we refer to as “the operational calculus method”, has the following two limitations: First,it does not provide any information about vectors for which the established inequalities become equalities (a matrix opti- mization problem). Second, the operational calculus method is futile in extending Kantorovich-type inequalities to operators on infinite dimensional Hilbert spaces. See [

It was in his investigation on problems of antieigenvalue theory that the author discovered and proved the Two Nonzero Component Lemma (see [

Lemma 1 (The Two Nonzero Component Lemma) Let

Let

be a function from

minimizing vectors for the function

on the convex set

What make the lemma possible are the following two facts: First, the fact that the set

is convex. Second, a special property that the functions

involved possess. If we set

then all restrictions

of

obtained by setting one component

call functions with this property self-reducing functions. Please note that TNCL is valid for both finite and in- finite variable cases. Let us look at an example of a self-reducing function where there are only a finite number of variables

where

Let

then we have

which has the same algebraic form as

Indeed, for any

obtained by setting an arbitrary set of

braic form as

Obviously, not all functions have this property. For instance, for the function

in the statement of TNCL above are finite or infinite linear combinations of

originally stated this way because when we deal with a matrix or operator optimization problem each

either a finite or infinite linear combination of variables

Example 2 In Theorem 1 of [

The unit vectors

is attained are called stationary values for (14). In Theorem 1 of [

In this case

set of eigenvalues of

In this section we will show how a generalization of TNCL can be formulated. In the proof of the TNCL in [

is a convex set and the function

is a self-reducing function. A function

can be a self-reducing function without being composed of linear combinations of the form

Example 3 Consider the function

This function is self-reducing but is not a composition of linear combinations. A close look at our arguments in [

set

We state the lemma for the case that the number of variables in finite (a case which occurs in many applied problems) but the arguments used in [

Lemma 4 If the function

is a positive self-reducing function on the convex set

then the minimizing vectors

have at most two nonzero components.

We call the lemma stated above the General Two Nonzero Component Lemma or GTNCL for short.

Remark 5 We can also use TNCL and GTNCL to find the maximum of a positive self-reducing function on (19). To see this please note that if

is a positive self-reducing function so is

and maximum of (20) on (19) is the reciprocal of the minimum of (21) on (19).

A general resource allocation problem is stated as

which can be converted to

In the following sections we will use GTNCL to compute a closed form answer for the distribution of the search effort problem.

This problem is formulated as

where

the conditional probability of detecting the object at position

then the distribution of the search effort problem will be transformed into

Theorem 6 The minimum of

subject to

is either

for some

for a a pair of

Proof. Since

is a self -reducing function, the GTNCL can be used to find the minimum of this function subject to

Since

subject to

Expression (28) can be simplified to

Substituting

If we differentiate (31) with respect to

If we solve (32) for

Substituting

If we substitute (33) and (34) in (30) we have

The last expression is equivalent to

Please note that the derivative of the function

with respect to

which is positive for

Although the GTNCL states two components

Theorem 7 Suppose the probabilities

Then the minimum of

subject to

is

Proof. Assume

Since

Furthermore, in (25) assume

as

Note that

then

Since

This shows that

the function

has its minimum at

and in this case the minimum of the objective function is

for some

for

Since both TNCL and GTNCL are valid for infinite number of variables, these techniques can be used to solve resource allocatoin problems involving an infinite number of variables as well. For example, in the distri- bution of the search effort problem we can assume the search is for an object on the plane that can be potentially detected at an infinite set of locations (such as points with integer

There are other resource allocation problems that we are able to tackle with GTNCL, One of these problems is the problem of optimal portfolio selection. One model for this general problem is formulated as finding the maximum of

(see [

If the correlation coefficients between

Notice that (44) is a self-reducing function and one can again apply the GTNCL to find a maximum value for it.

The problems of distribution of search effort and optimal portfolio selection are both examples of separable resource allocation problems. A separable resource allocation problem is a problem where we want to minimize or maximize

where each

such a problem if

allocation problems including optimal sample allocation in stratified sampling, and production planning (see [

Remark 8 In a broader sense, each Kantorovich-type matrix optimization problem such as the one in Example 2 can be regarded as a resource allocation problem where our resource is just the set of pure numbers on the interval

Each

optimization problem (15). Indeed nonzero components of a minimizing vector

Remark 9. The author is not aware of any other theorem that provides closed form answers for resource allocation problems. The results we obtain might be of interest for instance in signal analysis where one needs to minimize the resource spent finding a signal that is probable to detected at a certain location. Computer models are used for solving such problems and it is interesting to investigate how consistent the results of computer models are with our results here. Also, many theories in portfolio selection suggest that diversification maximizes the profit. At the first glance this might sound inconsistent with the results one might obtain using the two nonzero theorem. However, we have to remember that the over theory also ensures diversification increases profit. If the profit is maximized for one or two securities, then the more the number of securities, the more pairs of securities we have.