A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

  title={A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space},
  author={Frank Deutsch and Wu Li and Joseph D. Ward},
  journal={Journal of Approximation Theory},
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 anyx?Xfrom the setK?C?A?1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb?Y. 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… 
A Global Approach to Nonlinearly Constrained Best Approximation
ABSTRACT In this paper, we study the problem of whether the best approximation to any x in a Hilbert space X from the set where C is a closed convex subset of X, S is a closed convex cone that does
Limiting epsilon-subgradient characterizations of constrained best approximation
Duality for optimization and best approximation over finite intersections
Recently Deutsch, Li and Swetits [2] have studied, in Hilbert space, a dual problem (Qm ) to the primal problem (P) of minimization of a special class of convex functions f over the intersection of m
Best Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space, Weak Slater Conditions, and the Strong Conical Hull Intersection Property
The "strong conical hull intersection property" (strong CHIP), which was introduced by us in 1997, is shown to be both necessary and sufficient for the following "perturbation property" to hold: determining the best approximation from the set $K$ to any point is equivalent to the (generally easier) problem of determining the worst approximation.
Regularities and their relations to error bounds
It is shown that a proper lower semicontinuous function f on X has a Lipschitz error bound if and only if the pair {epi(f),X×{0}} of sets in the product space X×ℝ is linearly regular (resp., regular).
On best restricted range approximation in continuous complex-valued function spaces
A new projection method for finding the closest point in the intersection of convex sets
This paper presents a new iterative projection method, termed AAMR for averaged alternating modified reflections, which can be viewed as an adequate modification of the Douglas–Rachford method that yields a solution to the best approximation problem.


Constrained best approximation in Hilbert space III. Applications ton-convex functions
This paper continues the study of best approximation in a Hilbert spaceX from a subsetK which is the intersection of a closed convex coneC and a closed linear variety, with special emphasis on
Constrained best approximation in Hilbert space
A simple geometric property called the “conical hull intersection property” is introduced that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation.
Best interpolation with convex constraints
Abstract A characterization of any solution to the minimization problem min{||x − z|| : x ∈ K ≔ C ∩ A −1 d} is given, where A is a continuous linear map from a real Banach space X to a locally convex
Interpolation from a Convex Subset of Hilbert Space: A Survey of Some Recent Results
Let C be a closed convex subset of the Hilbert space H, A a bounded linear operator from H to a finite-dimensional Hilbert space Y, b ∈ Y, and K(b) = C ∩A -1(b). A survey is given of recent results
Smoothing and Interpolation in a Convex Subset of a Hilbert Space
Interpolation and smoothing subject to convex constraints is considered. We clarify instances in a Hilbert space when the problem of finding the least norm solution to these problems can be separated
A Method for Finding Projections onto the Intersection of Convex Sets in Hilbert Spaces
Many problems require the ability to find least squares projections onto convex regions. Here it is shown that if the constraint region can be expressed as a finite intersection of simpler convex
Approximation Theory, Wavelets and Applications
Preface. A Class of Interpolating Positive Linear Operators: Theoretical and Computational Aspects G. Allasia. Quasi-Interpolation E.W. Cheney. Approximation and Interpolation on Spheres E.W. Cheney.
A nonlinear equation for linear programming
It is shown thatx* is the optimal solution of (P), of minimal norm, if and only if there exists anR > 0 such that, for eachr ≥ R, the authors havex* = (rc − Atλr)+.
Gauss-seidel method for least-distance problems
This paper reformulates the least-distance problems with bounded inequality constraints as an unconstrained convex minimization problem, which is equivalent to a system of piecewise linear equationsA(a+ATy)cd=b and proves that the Gauss-Seidel method has a linear convergence rate.