Stability of best rational Chebyshev approximation
The authors study best Chebyshev approximation from the standpoint of optimization. Consider \((y,z)\in {\mathbb{R}}^ n\times {\mathbb{R}}\) and pose the problem minimize \(p(v,z)=z\), subject to (i) \(v\in U\), (ii) for all (\(\eta,s)\in T:\) \(\eta \frac{<B_{s,v}>}{<C_{s,v}>}-\gamma (\eta,s)z\leq \eta x(s),\) where \(S\neq \emptyset\) is compact Hausdorff space, \(B,C: S\to R^ n\) are continuous functions such that the open convex set \(U=\cap_{s\in S}\{v\in R^ n| <C(s),v>>0\}\) is nonempty, \(\gamma: T=\{-1,1\}\times S\to R\) is a nonnegative continuous function such that \(\gamma(-1,s)+ \gamma(1,s)>0\) for all \(s\in S\) and \(x: S\to R\) is continuous. The paper then goes on introducing feasible sets, the minimal value for \(p(v,z)\), the solutions attaining this value, the solutions that can be written as a generalized rational function and attain this minimal value and the class of problems \((B,C,\gamma,x)\) for which the solvability set is nonempty and a condition for a problem to belong to this class (the Slater condition). Then a number of results on lower/upper (semi)continuity of the set theoretic mappings connected with the feasible sets, minimal values and minimal solution sets are given. The style of the paper is rather compact but - after mastering the definitions and concepts that play a role - it is very interesting and quite readable.
scientific article; zbMATH DE number 4200864
scientific article