Effects of Different Objective Functions in Inequality Constrained and Rank-Deficient Least-Squares Problems

Abstract

Rank-deficient estimation problems often occur in geodesy due to linear dependencies or underdetermined systems. Well-known examples are the adjustment of a free geodetic network or a finite element approximation with data gaps. If additional knowledge about the parameters is given in form of inequalities (e.g., non-negativity), a rank-deficient and inequality constrained adjustment problem has to be solved.

In Roese-Koerner and Schuh (J Geodesy, doi:10.1007/s00190-014-0692-1) we proposed a framework for the rigorous computation of a general solution for rank-deficient and inequality constrained least-squares problems. If the constraints do not resolve the manifold of solutions, a second minimization is performed in the nullspace of the design matrix. This can be thought of as a kind of pseudoinverse, which takes the inequality constraints into account.

In this contribution, the proposed framework is reviewed and the effect of different objective functions in the nullspace optimization step is examined. This enables us to aim for special properties of the solution like sparsity (L ¹ norm) or minimal maximal errors (L ^∞ norm). In a case study our findings are applied to two applications: a simple bivariate example to gain insight into the behavior of the algorithm and an engineering problem with strict tolerances to show its potential for classic geodetic tasks.