A novel globally convergent maximizer for the multivariate carrier-phase integer ambiguity function

Current theory of integer inference, which is key to many carrier-phase driven observing systems, consists of a rich variety of different estimation principles, each with their own optimality and statistical properties. The various estimators can be classified into different classes of estimators, of which the integer estimation class is the smallest and the integer equivariant class the largest. Although the estimation theory for mixed integer models has matured significantly, there are still some important identifiable open unsolved problems. One of those concerns the way in which in practice the integer-equivariant baseline maximizer of the carrier-phase integer ambiguity function is resolved. Most of the methods employed in practice use rather ad hoc, brute-force grid search techniques, whereby proper considerations of the intrinsic properties of the objective function are lacking. As a result none of the available techniques have a demonstrated proven guarantee of global convergence. In this contribution we will present a novel algorithm for the numerical maximization of the multivariate carrier-phase integer ambiguity function. Our proposed method, which has finite termination with a guaranteed user-defined tolerance, is developed from combining the branch-and-bound principle with the projected-gradient-descent methodology, for which a special continuous differentiable convex-relaxation of the critical elements of the ambiguity objective function is constructed. The methodology of these three constituents is described in an integrated manner and numerical results are provided to illustrate the theory.