Duality and Lipschitzian selections in best approximation from nonconvex cones
The author's abstract states: ``Duality relationships in finding a best approximation from a nonconvex cone in a normed linear space in general, and in the space of bounded functions in particular, are investigated. The cone and the dual problems are defined in terms of positively homogeneous super-additive functionals on the space. Conditions are developed on the cone so that the duality gap between a pair of primal and dual problems does not exist. In addition, Lipschitz continuous selections of the metric projection are identified. The result are specialized to a convex cone. Applications are indicated to approximation problems.'' The paper is concerned with spaces \(B\) of real functions \(f\) on a set \(S\) which are such that \(wf\) is bounded, where \(w\) is a fixed positive weight function, and best approximation in \(B\) from a cone \(K\). A space \(B\) is normed by \(\| f\|=\sup| f| w\). The general results obtained are applied to the following examples: (4.1) \(S\) a convex subset of a vector space and \(K\) the set of all quasi-convex functions in \(B\); (4.2) \(S\) a convex subset of a vector space and \(K\) the set of all convex functions in \(B\); (4.3) \(S\) a proper open interval and \(K\) the set of all sub-additive functions in \(B\); and (5.1) certain generalised problems of monotone approximation with constraints. The paper is closely related to earlier work of the author [e.g., ibid. 55, No. 3, 326-336 (1988; Zbl 0699.41018) and ibid. 56, No. 2, 217-224 (1989; Zbl 0673.41027)].
scientific article; zbMATH DE number 8513
scientific article