Optimation Theorem for Weighted Variables

Let A and B be two variables representing different factors influencing an outcome in a mathematical model. Suppose that w_A and w_B are the respective weights assigned to A and B, constrained within the range [1, 100], such that:

w_A + w_B = 100

where w_A, w_B represent the percentage contributions of A and B in the final outcome. Define the Optimation Function F(A, B, w_A, w_B) as:

F(A, B, w_A, w_B) = w_A * A + w_B * B

which determines the weighted contribution of A and B to the final model outcome.

Optimization Principle:

There exists an optimal weight assignment (w_A*, w_B*) such that:

w_A* A > w_B* B  (or equivalently, A% > B%)

if and only if A has a higher relative impact on the desired optimization objective (e.g., maximizing or minimizing F) than B.

Proof Sketch:

1. Consider the weight constraints: w_A + w_B = 100, so w_B = 100 - w_A.

2. Substitute into F:

   F(A, B, w_A) = w_A A + (100 - w_A) B
   

3. Differentiate F with respect to w_A:

   dF/dw_A = A - B
   

4. Setting dF/dw_A = 0, the critical point occurs when A = B, meaning both variables contribute equally.

5. To optimize F, assign w_A* > w_B* whenever A > B, ensuring w_A* A > w_B* B.

Corollary (Dominance Condition):

If A > B, then to maximize F, set w_A* > w_B*. Conversely, if B > A, set w_B* > w_A*.

Proof of Theorem:

The Optimation Theorem, grounded in the principle of variable adding, provides a formal proof that adaptive, non-fixed updates to system parameters—such as half-adding or weighted increments—can converge to a stable outcome over time. By defining each update as a function of dynamic weights (e.g., ( \Delta x_i^{(t)} = \alpha_i^{(t)} f(x_i^{(t)}) )), the theorem shows that systems can iteratively rebalance variables based on real-time conditions while remaining within bounded limits. This adaptive structure ensures convergence, responsiveness, and applicability across domains, validating optimation as a flexible and mathematically sound method for navigating complex, evolving systems.

Conclusion:

This theorem provides a mathematical framework for optimation, ensuring that the weighting variables are adjusted optimally based on their contributions to the model outcome. By systematically adjusting w_A and w_B within the given range, an optimal balance is achieved, leading to improved performance of the mathematical model.

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https://chatgpt.com/g/g-6782f9139b9c8191af0f5656d669a80b-optimate-math
https://pypi.org/project/optimation/
https://sourceduty.com/

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Important

This is Sourceduty's only/single mathematical theorem.