Mixed Integer Linear Models

Space geodesy has brought profound convenience to our daily life with positioning-based products on one hand and provided a challenging opportunity to contribute fundamentally to mathematics and statistics on the other hand. The use of carrier phase observables has given rise to new observation models we have never encountered in any course/lecture of statistics and/or adjustment theory. This chapter is to provide a tutorial on mixed integer linear models. We first classify real-valued and mixed integer linear models and, then, accordingly define the corresponding conventional and mixed integer least squares (ILS) problems. Since the weighted ILS problem or the closest point problem is NP-hard, we start with suboptimal integer solutions. Reduction and/or decorrelation methods are briefly reviewed for practical applications. Integer unknown parameters are then exactly solved under a general framework of integer programming (aided with decorrelation and/or reduction methods) and represented/estimated from the statistical point of view. As an indispensable and fundamental element of integer least squares estimator, the Voronoi cell is shown to be extremely complex, both computationally and in shape, and has to be bounded with figures of simple shape. As a direct application, we obtain lower and upper probabilistic bounds for the probability with which the integers are correctly estimated. Finally, we briefly discuss an integer hypothesis testing problem.