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

• Authors: M. J. D. Powell
• Venue: Computational Optimization and Applications - Special issue on computational optimization—a tribute to Olvi Mangasarian, part II
• Year: 1999

Let \underline{s}(t), 0 ≤ t ≤ T, be a smooth curve and let \underline{x}_i, i = 1, 2, \ldots, n, be a sequence of points in two dimensions. An algorithm is given that calculates the parameters t_i, i = 1, 2, \ldots, n, that minimize the function max{‖\underline{x}_i − \underline{s}(t_i) ‖_2 : i = 1, 2, \ldots, n } subject to the constraints 0 ≤ t_1 ≤ t_2 ≤ \cdots ≤ t_n ≤ T. Further, the final value of the objective function is bestlexicographically, when the distances ‖\underline{x}_i − \underline{s}(t_i)‖_2,i = 1, 2, \ldots, n, are sorted into decreasing order. Thealgorithm finds the global solution to this calculation. Usually themagnitude of the total work is only about n when the number of datapoints is large. The efficiency comes from techniques that use boundson the final values of the parameters to split the original probleminto calculations that have fewer variables. The splitting techniquesare analysed, the algorithm is described, and some numerical resultsare presented and discussed.