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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.

KillaKem
  • 695
  • What have you tried? What exactly are you having trouble with? Also, are your arrays matrices or vectors? In the case that they are vectors, and you're using the dot product, it is much simpler than otherwise. – Steve Heim Mar 30 '14 at 19:40

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