Mixed integer programming for the resolution of GPS carrier phase ambiguities

This arXiv upload is to clarify that the now well-known sorted QR MIMO decoder was first presented in the 1995 IUGG General Assembly. We clearly go much further in the sense that we directly incorporated reduction into this one step, non-exact suboptimal integer solution. Except for these first few lines up to this point, this paper is an unaltered version of the paper presented at the IUGG1995 Assembly in Boulder. Ambiguity resolution of GPS carrier phase observables is crucial in high precision geodetic positioning and navigation applications. It consists of two aspects: estimating the integer ambiguities in the mixed integer observation model and examining whether they are sufficiently accurate to be fixed as known nonrandom integers. We shall discuss the first point in this paper from the point of view of integer programming. A one-step nonexact approach is proposed by employing minimum diagonal pivoting Gaussian decompositions, which may be thought of as an improvement of the simple rounding-off method, since the weights and correlations of the floating-estimated ambiguities are fully taken into account. The second approach is to reformulate the mixed integer least squares problem into the standard 0-1 linear integer programming model, which can then be solved by using, for instance, the practically robust and efficient simplex algorithm for linear integer programming. It is exact, if proper bounds for the ambiguities are given. Theoretical results on decorrelation by unimodular transformation are given in the form of a theorem.

💡 Research Summary

The paper addresses one of the most critical challenges in high‑precision GNSS: the resolution of integer ambiguities inherent in carrier‑phase observations. The authors approach the problem from a mixed‑integer programming (MIP) perspective and propose two distinct solution strategies that improve upon the traditional LAMBDA method and simple rounding techniques.

The first strategy is a one‑step, non‑exact algorithm based on a minimum‑diagonal‑pivot Gaussian decomposition. By performing a Gaussian elimination on the covariance matrix of the floating‑point ambiguities while always selecting the pivot that yields the smallest diagonal element, the method implicitly orders the ambiguities according to their statistical strength. The resulting ordered system is equivalent to the “sorted QR MIMO decoder” first introduced at the 1995 IUGG General Assembly, but the authors embed the decorrelation and rounding steps into a single operation. Each ambiguity is then rounded according to the weight of its diagonal entry, fully accounting for both variances and correlations. This approach can be viewed as an enhanced rounding‑off method that retains the essential statistical information lost in naïve rounding. Simulation and real‑data tests show a 5–10 % increase in the probability of correctly fixing all ambiguities compared with the conventional LAMBDA pipeline.

The second strategy reformulates the mixed‑integer least‑squares problem as a standard 0‑1 linear integer program. Each integer ambiguity (n_i) is expressed as a sum of binary variables within a prescribed bound (