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 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. 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, and give graphical illustrations.

翻译：我们考虑一个在热带代数框架下提出的离散最佳逼近问题，该问题涉及幂等运算代数系统的理论与应用。给定未知函数的输入输出样本集，问题在于构造一个广义热带皮瑟多项式，使其在热带距离函数意义下最佳逼近该函数。近似多项式的构建需要同时确定多项式各项的未知系数和指数。为解决该逼近问题，我们首先将问题转化为一个关于未知系数向量的方程，该方程由矩阵给出，其元素由未知指数参数化。我们推导出方程的最佳近似解，该解同时给出系数向量和由指数参数化的逼近误差。通过将逼近误差最小化问题转化为对有限集所有划分的指数函数最小化问题，求得指数的最优值。我们采用基于凝聚聚类技术的计算过程，在最大-正代数框架下解决该最小化问题。该解法被推广至最大代数框架下寻找多项式最优指数的最小化问题。所得结果被应用于开发实函数分段线性逼近和分段皮瑟多项式离散最佳切比雪夫逼近问题的新解法。我们讨论了所提解法的计算复杂度，并估算了计算时间上限。通过展示使用最大-正代数和最大代数求解的逼近问题实例，并给出图示说明。