Optimization Models for Interpolating Overlapping Interval Data

Abstract

We consider a general robust spline interpolating computation problem where the intervals representing the ranges in positions of knots after clustering can be overlapped. We discuss and analyze the similarities and differences of different overlapping situations, propose two definitions of overlapping types for the simple overlapping situations: PIO (partial interval overlapping) and WIO (whole interval overlapping). An optimization model called PHOM-S (position-height optimization model with a single overlapping interval) is presented and formulated as a convex semi-infinite optimization problem which is capable of solving the two overlapping types mentioned. Then a brief discussion is given on how to build PHOM-M (position-height optimization model with multiple overlapping intervals) based on PHOM-S for solving general simple overlapping problems with multiple overlapping intervals. Also, simplification of a finite optimization model called PHOM-S-3 using the cubic polynomial spline is given for the applicability of computation. A numerical example is presented to show the feasibility and effectiveness of the proposed robust model PHOM-S-3.