Optimality Conditions for Multivariate Chebyshev Approximation: A Survey

TL;DR

This survey reviews various optimality conditions for multivariate Chebyshev approximation, highlighting theoretical foundations, relationships, and practical applications in multidimensional polynomial approximation problems.

Contribution

It compiles and analyzes existing optimality conditions for multivariate Chebyshev approximation, including new insights from convex analysis and practical numerical applications.

Findings

Comparison of optimality conditions including Kirchberger, Kolmogorov, Rivlin-Shapiro, Bartelt, and Smarzewsky.

Relationships between optimality conditions and convex analysis concepts like subdifferential and directional derivatives.

Numerical demonstrations on multidimensional approximation problems, including Runge function and robot inverse model.

Abstract

Uniform polynomial approximation, also called minimax approximation or Chebyshev approximation, consists in searching polynomial approximation that minimizes the worst case error. Optimality conditions for the uniform approximation of univariate functions defined in an interval are governed by the equioscillation theorem, which is also a key ingredient in algorithms for computing best uniform approximation, like Remez's algorithm and the two phase method. Multivariate polynomial approximation is more complicated, and several optimality conditions for uniform multivariate polynomial approximation generalize the equioscillation theorem. We review these conditions, including, from oldest to newest, Kirchberger's kernel condition, Kolmogorov criteria, Rivlin and Shapiro's annihilating measures. An emphasis is given to conditions for strong optimality, which has some strong theoretical and practical importance, including Bartelt's and Smarzewsky's conditions. Optimality conditions related to more general relative Chebyshev centers are also presented, including Tanimoto's and Levis et al.'s conditions. In a second step, conditions obtained by standard convex analysis, subdifferential and directional derivative, applied to uniform approximation are formulated. Their relationship to previous conditions is investigated, providing sometimes enlightening interpretations of the laters, e.g., relating Kolmogorov criterion with directional derivative, and strong uniqueness with sharp minimizers. Finally, numerical applications of the two-step approach to three uniform approximation problems are presented, namely the approximation of the multidimensional Runge function, the approximation of the two dimensional inverse model of the DexTAR parallel robot, and the approximation problem consisting in minimizing the sum of both the polynomial approximation error and the polynomial evaluation error in Horner form.