Hi. I have a problem I am hoping you all can shed some light on.(adsbygoogle = window.adsbygoogle || []).push({});

I have N entities, O, each described by N values - a weight W and N-1 similarity coefficients to the other N-1 entities. I guess we can represent O_{i}as (W_{i}, S_{ij, j=(1,2,...,N, i!=j})(?).

Given an integer M and M < N I need to maximize the following function:

Sum_{i}(W_{i}/Sum_{j}(S_{ij, j=(1,2,...,N), i!=j}))

where i and j in the sums are constrained to the unique combinations of the integers 1 to N of size M.

As an example let N=4, M=3 there are then 4 unique combinations of the numbers 1,2,3,4 of size 3: (1,2,3), (1,2,4), (1,3,4) and (2,3,4). The values for the function are therefore:

W_{1}/(S_{1,2}+S_{1,3})+W_{2}/(S_{2,1}+S_{2,3})+W_{3}/(S_{3,1}+S_{3,2})

W_{1}/(S_{1,2}+S_{1,4})+W_{2}/(S_{2,1}+S_{2,4})+W_{4}/(S_{4,1}+S_{4,2})

W_{1}/(S_{1,3}+S_{1,4})+W_{3}/(S_{3,1}+S_{3,4})+W_{4}/(S_{4,1}+S_{4,3})

W_{2}/(S_{2,3}+S_{2,4})+W_{3}/(S_{3,2}+S_{3,4})+W_{4}/(S_{4,2}+S_{4,3}).

For small N and M I can enumerate the combinations and calculate the maximum sum but my problem has N~50 and M~5.

Any suggestions or thoughts of an akgorithm to calulate the max? I hope I have made it clear.

Thanks in advance.

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# Optimization problem

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