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 reduced to a 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 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表述的离散最佳逼近问题，该代数涉及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，问题在于构造一个广义热带普伊瑟多项式，使其在热带距离函数意义下最佳逼近该函数。逼近多项式的构建需同时求解多项式中每个单项式的未知系数与指数。为解决该逼近问题，我们首先将问题归结为一个关于未知系数向量的方程，该方程由矩阵给出，矩阵元素由未知指数参数化。我们推导出方程的最佳逼近解，该解同时给出系数向量和由指数参数化的逼近误差。通过最小化逼近误差来确定指数的最优值，该过程可转化为在有限集合的所有划分上对指数函数的极小化问题。我们利用基于凝聚聚类技术的计算流程，在最大-加法代数（其中加法定义为取最大值，乘法定义为算术加法）框架下求解该极小化问题。该解法可推广至最大代数（其中加法定义为取最大值）中寻找多项式最优指数的极小化问题。将所得结果应用于实函数分段线性函数和分段普伊瑟多项式离散最佳逼近的传统问题，提出新型解法。我们讨论了所提解法的计算复杂度并估算了计算时间的上界，展示了以最大-加法代数和最大代数求解逼近问题的示例，并给出图形说明。