Unified Least-Squares Formulation of a Linear Model with Hard Constraints

A. R. Amiri-Simkooei*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


Hard constraints can play an important role in obtaining a unique, stable, precise, and well-conditioned least-squares solution in many geomatics applications. This paper presents a unified formulation for the least-squares solution applicable to full-rank, ill-conditioned, and rank-deficient linear models subject to hard constraints having regularizing effects. The formulation is unified because, in particular, it can handle all kinds of linear models in which the constraints can be extra, redundant, or minimal. Previous work usually considered different formulations when incorporating the extra constraints compared with the minimum ones. The unified formulation is thus proposed regardless of whether the constraints are minimal, redundant, or extra and whether the linear model is of full rank, ill conditioned, or rank deficient. The proposed formulation can in particular be applied to two commonly used problems: (1) ill-posed problems in which the hard constraints have regularization effect; and (2) a rank-deficient geodetic network in which the rank deficiency is caused by the lack of the datum definition - free network adjustment, for example. Four examples are employed to investigate the performance of the proposed formulation. The first example is a leveling network. The second example handles an ill-posed least-squares problem. The third example applies the proposed formulation to the least-squares cubic approximation problem. The fourth example considers estimating parameters in a three-dimensional (3D) similarity coordinate transformation problem. All examples confirm the efficacy of the unified formulation.

Original languageEnglish
Article number04019012
Number of pages11
JournalJournal of Surveying Engineering
Issue number4
Publication statusPublished - 1 Nov 2019


  • Least-squares method
  • Linear hard constraints
  • Minimum constraints
  • Three-dimensional (3D) affine transformation

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