Globally optimal univariate spline approximations

Robert Mohr, Maximilian Coblenz*, Peter Kirst

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


We revisit the problem of computing optimal spline approximations for univariate least-squares splines from a combinatorial optimization perspective. In contrast to most approaches from the literature we aim at globally optimal coefficients as well as a globally optimal placement of a fixed number of knots for a discrete variant of this problem. To achieve this, two different possibilities are developed. The first approach that we present is the formulation of the problem as a mixed-integer quadratically constrained problem, which can be solved using commercial optimization solvers. The second method that we propose is a branch-and-bound algorithm tailored specifically to the combinatorial formulation. We compare our algorithmic approaches empirically on both, real and synthetic curve fitting data sets from the literature. The numerical experiments show that our approach to tackle the least-squares spline approximation problem with free knots is able to compute solutions to problems of realistic sizes within reasonable computing times.

Original languageEnglish
Pages (from-to)409-439
Number of pages31
JournalComputational Optimization and Applications
Issue number2
Publication statusPublished - 28 Feb 2023


  • Branch-and-bound
  • Curve fitting
  • Global optimization
  • Least-squares spline approximation
  • Spline approximation


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