We consider a discrete best approximation problem formulated in the framework of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. Given a set of samples of input and output of an unknown function, the problem is to construct a generalized tropical Puiseux polynomial that best approximates the function in the sense of a tropical distance function. The construction of an approximate polynomial involves the evaluation of both unknown coefficient and exponent of each monomial in the polynomial. To solve the approximation problem, we first reduce the problem to an equation in unknown vector of coefficients, which is given by a matrix with entries parameterized by unknown exponents. We derive a best approximate solution of the equation, which yields both vector of coefficients and approximation error parameterized by the exponents. Optimal values of exponents are found by minimization of the approximation error, which is transformed into minimization of a function of exponents over all partitions of a finite set. We solve this minimization problem in terms of max-plus algebra (where addition is defined as maximum and multiplication as arithmetic addition) by using a computational procedure based on the agglomerative clustering technique. This solution is extended to the minimization problem of finding optimal exponents in the polynomial in terms of max-algebra (where addition is defined as maximum). The results obtained are applied to develop new solutions for conventional problems of discrete best Chebyshev approximation of real functions by piecewise linear functions and piecewise Puiseux polynomials. We discuss computational complexity of the proposed solution and estimate upper bounds on the computational time. We demonstrate examples of approximation problems solved in terms of max-plus and max-algebra.

翻译：我们考虑在热带代数框架下提出的离散最佳逼近问题，该框架研究具有幂等运算的代数系统的理论与应用。给定未知函数的输入输出样本集，该问题旨在构造一个广义热带Puiseux多项式，使其在热带距离函数的意义下最佳逼近该函数。近似多项式的构造涉及对多项式中每个单项式的未知系数和指数的求解。为解决该逼近问题，我们首先将问题转化为关于未知系数向量的方程，该方程的矩阵元素由未知指数参数化。我们推导出方程的最佳近似解，该解给出了由指数参数化的系数向量和逼近误差。通过最小化逼近误差获得指数的最优值，该最小化问题被转化为在有限集的所有划分上对指数函数的最小化。我们基于凝聚聚类技术，利用极大加代数（其中加法定义为取最大值，乘法定义为算术加法）的计算程序求解该最小化问题。该解法被推广至在极大代数（其中加法定义为取最大值）框架下寻找多项式中最优指数的最小化问题。所得结果被应用于开发新解法，以解决用分段线性函数和分段Puiseux多项式对实函数进行离散最佳切比雪夫逼近的传统问题。我们讨论了所提解法的计算复杂度，并估算了计算时间的上界。最后展示了在极大加代数和极大代数框架下求解的逼近问题实例。