Abstract

When studying complex optimization and control problems for systems described by polynomial and analytic functions, there is often a need to use necessary and sufficient optimality conditions. Moreover, if the known conditions turn out to be inapplicable, it is required to develop as subtle conditions as possible. This problem is studied in this article. The necessary and sufficient conditions for a local extremum are formulated for polynomials and power series. With a small number of variables, these conditions can be tested using practically implemented algorithms. The main ideas of the proposed methods involve using the Newton polytope for a polynomial (power series) and the expansion of a polynomial (power series) into a sum of quasi-homogeneous polynomial forms. The obtained results provide the practically applicable methods and algorithms necessary for solving complex problems of optimization and control of systems, which are described by polynomial and analytical functions. Specific examples of tasks in which the proposed technique can be used are given.

中文翻译：

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多项式和解析函数描述的系统优化复杂问题中极值的充分必要条件

摘要

在研究由多项式和解析函数描述的系统的复杂优化和控制问题时，通常需要使用必要且充分的最优性条件。此外，如果已知的条件不适用，则需要制定尽可能微妙的条件。本文对这个问题进行了研究。局部极值的充分必要条件是针对多项式和幂级数制定的。通过少量变量，可以使用实际实现的算法来测试这些条件。所提出方法的主要思想涉及使用牛顿多胞形来表示多项式（幂级数）以及将多项式（幂级数）展开为准齐次多项式形式的和。所获得的结果为解决系统优化和控制的复杂问题提供了实际适用的方法和算法，这些问题由多项式和解析函数描述。给出了可以使用所提出的技术的任务的具体示例。