A dual approach to constrained interpolation from a convex subset of hilbert space

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyxXfrom the setKCA-1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andbY. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y) - the best approximationto a certain perturbationx+A*yofx - from the convexsetCor from a certain convex extremal subsetCbofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA-1(b) for which this canalways be done. Finally, in many cases, the best approximationPC(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence 1997 Academic Press.

A dual approach to constrained interpolation from a convex subset of hilbert space Academic Article