An Optimal Way of Moving a Sequence of Points onto a Curve in Two Dimensions

Let s(t), 0 t T, be a smooth curve and let a sequence of points in two dimensions. An algorithm is given that calculates to the constraints 0 t 1 t 2 t n T. Further, the nal value of the objective function is best lexicographically, when the distances kx i ?s(t i)k 2 , i=1; 2; : : : ; n, are sorted into decreasing order. The algorithm nds the global solution to this calculation. Usually the magnitude of the total work is only about n when the number of data points is large. The eeciency comes from techniques that use bounds on the nal values of the parameters to split the original problem into calculations that have fewer variables. The splitting techniques are analysed, the algorithm is described, and some numerical results are presented and discussed. 1 This paper is dedicated to Olvi Mangasarian in celebration of his 65th birthday. The author is one of many people in optimization who appreciate greatly the quality of his academic work, his friendship, and his willingness to discuss research.