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Equation (A7) leads straight away towards the optimal value of ? * appearing in Equation (8).

The presumption of analytical self-reliance signifies that the probability that is joint purpose of the elements of ?X? is:

(B1) f X; ? 1, ? 2, … ? m = ? i = 1 m ? i ag e – ? i = 1 m ? i x i (B1)

The likelihood that the arbitrary breakthrough will have attributes satisfying the conditions imposed by the minimal set is:

(B2) P r o b x 1 x 1, min, …. X n x n, min = 1 – F X min = e – ? i = 1 n ? i x i, min (B2)

The numerator for the anticipated energy in Equation (4) within the unique situation is:

(B3) ? x n, min ? ? ? x 1, min ? U X f X d x 1 ? d x n = ? x n, min ? ? ? x 1, min ? ? i = 1 n w i x i ? i = 1 n ? i ag ag ag e – ? i i ? i = 1 n d x i (B3)

The independence that is statistical of random factors in X enable the phrase in the right-hand part of Equation (B3) to solve because:

(B4) ? x n, min ? ? ? x 1, min ? U X f X d x 1 ? d x n = ? i = 1 n ? x i, min ? w i x i ? i e – ? i x i d x i (B4)

It really is a simple calculation to show that the definite integral within the summand in B4 is:

(B5) ? x i, min ? x i ? i e – ? i x i d x i = x i, min + 1 ? i ag e – ? i x i, min (B5)

Carrying out of the integration regarding the right-hand part of B4 and summing, we now have:

(B6) ? x n, min ? ? ? x 1, min ? U X f ( X ) d x 1 ? d x n = ? i = 1 n w i x i, min + 1 ? i ag e – ? i x i, min (B6)

The typical definitional equation for U ? given in (4) may be expressed in this situation as: (B7) U ? ? = ? i = 1 n w i x i, min + 1 ? i ag e – ? i x i, min ag ag e – ? i = 1 n ? i x i, min (B7)

Replacing Equation (B7) into ratio (9) to calculate 1 – ? leads straight to Equation (12).

Notes

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