Necessary and Sufficient Conditions for an Extremum in Complex Problems of Optimization of Systems Described by Polynomial and Analytic Functions

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.