An Approximate Solution of a GNSS Satellite Selection Problem Using Semidefinite Programming

Abstract

When processing multiple navigation satellite systems, including GPS, GLONASS, Galileo, Beidou, QZSS, the overall number of the pseudorange and carrier phase signals can exceed several tens. On the other hand, a much smaller number of them is usually sufficient to achieve necessary precision of positioning. Also, some parts of precise positioning algorithms, like carrier phase ambiguity resolution, are very sensitive to the problem dimension as they include the integer search. To reduce computational cost of positioning, the optimal choice of signals involved in computations should be performed. Optimization is constrained by a given number of satellite signals to be chosen for processing. This optimization problem falls into the class of binary optimization problems which are hard for precise solution. In this paper, we present approaches to an approximate solution of the optimal selection problem. After the linear relaxation of binary constraints, the relaxed problem is convex and can be transformed to semidefinite programming or second-order cone programming problems. The optimal solution of the relaxed problem can be considered as a lower bound of a combinatorial optimization problem. After rounding non-integer variables the approximate solution is obtained. As a result, two-sided bounds of the optimum are obtained. In practice, the approximate solution is very close to precise solution for most real world cases. Because the relaxed problem is convex, it can be solved efficiently.

This work was financially supported by the Russian Foundation for Basic Research, project 18-08-00531.