A dual approach to constrained interpolation from a convex subset of Hilbert space
Let \(X, Y\) be Hilbert spaces, \(Y\) finite dimensional and let \(A: X\rightarrow Y\) be a bounded linear operator. For the closed convex subset \(C\) of \(X\) and for a fixed \(b\in Y\) the authors consider the metric projection \(P_K\), where \(K= C \cap A^{-1}(b)\) and \(K \neq \emptyset\). The main purpose of this paper is to show that, under some assumptions, \(P_K(x)= P_C(x+ A^*y)\), with a convenable \(y \in Y\). The authors prove that such an equality is possible if and only if the pair \(\{C, A^{-1}(b)\}\) has ``the strong conical hull intersection property'' or, equivalently, if for every \(x \in X\) there exists \(y \in Y\) such that \(AP_C(x+A^*y)= b\). This is the case, particularly, when \(C\) is a polyhedron or \(b\) is in the relative interior of \(A(C)\). Moreover, they prove that there exists a certain convex extremal subset \(C_b\) of \(C\) such that \(C_b\cap A^{-1}(b)= C \cap A^{-1}(b)= K \) and such that, for each \(x\in X,P_K(x)= P_{C_b}(x+A^*y)\), with a convenable \(y= y(x) \in Y\). In a particular case, \(C_b\) and \(P_K(x)\) are effectively computed. For \(C\) a polyhedral set is described a linearly convergent algorithm, in order to obtain a sequence of approximants for \(P_K(x)\). In this case, for each \(x\in X, AP_C(x+A^*y)-b\) is the gradient of a convex (quadratic spline) function of variable \(y\).
algorithms for obtaining a best approximation element
geometry of convex sets in Hilbert spaces
scientific article; zbMATH DE number 1076160
scientific article