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Best interpolation in a strip II: Reduction to unconstrained convex optimization


In this paper, we study the problem of finding a real-valued function f on the interval [0, 1] with minimal L 2 norm of the second derivative that interpolates the points (t i, y i) and satisfies e(t) ≤ f(t) ≤ d(t) for t ∈ [0, 1]. The functions e and d are continuous in each interval (t i, t i+1) and at t 1 and t nbut may be discontinuous at t i. Based on an earlier paper by the first author [7] we characterize the solution in the case when e and d are linear in each interval (t i, t i+1). We present a method for the reduction of the problem to a convex finite-dimensional unconstrained minimization problem. When e and d are arbitrary continuous functions we approximate the problem by a sequence of finite-dimensional minimization problems and prove that the sequence of solutions to the approximating problems converges in the norm of W 2,2 to the solution of the original problem. Numerical examples are reported.

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The first author was supported by National Science Foundation Grant Number DMS 9404431. The second author was supported by a François-Xavier Bagnoud doctoral fellowship and by National Science Foundation Grant Number MSS 9114630.

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Dontchev, A.L., Kolmanovsky, I. Best interpolation in a strip II: Reduction to unconstrained convex optimization. Comput Optim Applic 5, 233–251 (1996). https://doi.org/10.1007/BF00248266

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  • constrained best approximation
  • splines
  • interpolation
  • convex programming