I would like to get a solution or receive guidance on how I can solve the optimisation problem below.
Let's say I have two arrays of length N , say A and B, and I want to find 2 coefficients $k_1$ and $k_2$ that will make distance between A and B minimum, ie find:
$$ k_1, k_2 \hspace{0.25cm} \text{such that distance}(k_1A,k_2B) \text{ is minimal} $$
where A = [$A_1$, $A_2$,..., $A_N$], B = [$B_1$,$B_2$,..$B_N$], and Distance($k_1A$, $k_2B$) is:
$$ \text{Distance}(k_1A,k_2B) = \sqrt{(k_1A_1 - k_2B_1)^2 + (k_1A_2 - k_2B_2)^2 + ... (k_1A_N - k_2B_N)^2} $$
Additional details about the nature of the arrays A and B that may restrict the solution set are stated below:
- Each Array has its elements in ascending order (e.g for $A$ : $A_{n+1} > A_{n}$ for $n = 1,...,N-1$)
- $1.5 < ( k_1 + k_2 ) < 2$
- $0.5 \leq \text{Min}(k_1,k_2) \leq 1$ and $1 \leq \text{Max}(k_1, k_2) \leq 1.5$
Any help will be appreciated.