We consider discrete best approximation problems in the setting of tropical algebra that is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input-output pairs of an unknown function defined on a tropical semifield, the problem is to determine an approximating rational function formed by two Puiseux polynomials as numerator and denominator. With specified numbers of monomials in both polynomials, the approximation aims at evaluating the exponent and coefficient for each monomial in the polynomials to fit the rational function to the data in the sense of a tropical distance function. To solve the problem, we transform it into approximation of a vector equation with unknown vectors on both sides, where one side corresponds to the numerator polynomial and the other side to the denominator. Each side involves a matrix with entries dependent on the unknown exponents, multiplied by the vector of unknown coefficients of monomials. We propose an algorithm that constructs a series of approximate solutions by alternately fixing one side of the equation to an already found result and leaving the other intact. Each equation obtained is approximated with respect to the vector of coefficients, which yields this vector and approximation error both parameterized by the exponents. The exponents are found by minimizing the error with an optimization procedure based on an agglomerative clustering technique. To illustrate, we present results for an approximation problem in terms of max-plus algebra (a real semifield with addition defined as maximum and multiplication as arithmetic addition), which corresponds to an ordinary problem of piecewise linear approximation of real functions. As our numerical experience shows, the proposed algorithm converges in a finite number of steps and provides a reasonable accurate solution to the problems considered.

翻译：本文在热带代数框架下研究离散最佳逼近问题，该代数体系以幂等运算为理论基础并应用于实际系统。给定定义于热带半域上的未知函数的一组输入-输出对，该问题旨在确定由两个Puiseux多项式（分别作为分子与分母）构成的逼近有理函数。在限定两多项式所含单项式数量的条件下，通过评估多项式中各单项式的指数与系数，使该有理函数在热带距离函数意义下拟合给定数据。为解决该问题，我们将其转化为具有双侧未知向量的向量方程逼近问题：其中一侧对应分子多项式，另一侧对应分母多项式。每侧均涉及一个矩阵（其元素依赖于未知指数）与单项式系数向量的乘积。我们提出一种算法，通过交替固定方程一侧为已得结果而保持另一侧不变，从而构建一系列逼近解。所得每个方程均针对系数向量进行逼近，由此得到由指数参数化的系数向量及逼近误差。指数通过基于凝聚聚类技术的优化程序最小化误差而确定。为具体说明，我们展示了基于最大-加代数（以最大值定义加法、算术加法定义乘法的实数半域）的逼近问题结果，该问题对应于实函数分段线性逼近的常规问题。数值实验表明，所提算法可在有限步内收敛，并为所研究问题提供具有合理精度的解。