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I have a quantity which is estimated from a number of noisy measurements. I know that the real underlying value must be some integer multiple of two quantities, e.g. $M = I_1C_1 + I_2C_2$ where $C_1$ and $C_2$ are some known (from theory) values and $I_1,I_2$ are the unknown integer multiples.

I'm casting this as a least squares minimisation problem, minimising over $I_1,I_2$ but I'm unsure of what minimisation technique is appropriate here? Is there some standard approach to this type of problem? I can get extreme values for the unknown integers by doing $ceil(M/C_i)$, so could then check every possible integer combination within that range. There are typically many such combinations though, so this would not be very efficient.

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I think this paper should be relevant to your problem:

Arash Hassibi and Stephen Boyd: Integer Parameter Estimation in Linear Models with Applications to GPS. IEEE Transactions on Signal Processing, Vol. 46, No. 11, November 1998

The authors state that the integer least-squares problem is NP-hard, but can be solved efficiently in many practical cases by the LLL algorithm.

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  • $\begingroup$ Thanks, this is indeed a GPS processing issue, so that reference is very relevant. $\endgroup$ – Bogdanovist May 29 '15 at 1:00

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