The problem of obtaining an optimal spline with free knots is tantamount to minimizing derivatives of a nonlinear differentiable function over a Banach space on a compact set. While the problem of data interpolation by quadratic splines has been accomplished, interpolation by splines of higher orders is far more challenging. In this paper, to overcome difficulties associated with the complexity of the interpolation problem, the interval over which data points are defined is discretized and continuous derivatives are replaced by their discrete counterparts. The $l_{\infty }$-norm used for maximum rth order curvature (a derivative of order r) is then linearized, and the problem to obtain a near-optimal spline becomes a linear programming (LP) problem, which is solved in polynomial time by using LP methods, e.g., by using the Simplex method implemented in modern software such as CPLEX. It is shown that, as the mesh of the discretization approaches zero, a resulting near-optimal spline approaches an optimal spline. Splines with the desired accuracy can be obtained by choosing an appropriately fine mesh of the discretization. By using cubic splines as an example, numerical results demonstrate that the linear programming (LP) formulation, resulting from the discretization of the interpolation problem, can be solved by linear solvers with high computational efficiency and the resulting spline provides a good approximation to the sought-for optimal spline.

中文翻译：

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使用线性规划通过带有自由结的近最佳样条进行数据插值

获得带有自由结的最佳样条的问题等同于最小化紧集上Banach空间上非线性可微函数的导数。尽管已经完成了通过二次样条进行数据插值的问题，但是通过更高阶的样条进行插值却更具挑战性。在本文中，为了克服与插值问题的复杂性相关的困难，对定义数据点的时间间隔进行离散化，并将连续的导数替换为其离散的对应项。这${升}_{\infty }$范数用于最大ř阶曲率（衍生物顺序的ř然后将其线性化，获得近似最佳样条的问题变为线性规划（LP）问题，可以通过使用LP方法（例如，使用在CPLEX等现代软件中实现的Simplex方法）在多项式时间内解决该问题。结果表明，随着离散网格的逼近为零，所得的近似最优样条曲线接近最优样条曲线。通过选择适当的离散网格，可以获得具有所需精度的样条曲线。通过以三次样条为例，数值结果表明，插值问题离散化后的线性规划（LP）公式可以由线性求解器以较高的计算效率求解，并且所得的样条可以很好地逼近所求-最佳样条。