On a finite kernel theorem for polynomial-type optimization problems and some of its applications

Abstract

Extremalization and optimization problems are considered as of utmost importance both in the past and in the present days. Thus, in the early beginning of infinitesimal calculus the determination of maxima and minima is one of the stimulating problems causing the creation of calculus and one of the successful applications which stimulates its rapid further development. However, the maxima and minima involved are all of local character leading to equations difficult to be solved, not to say the inherent logical difficulties involving necessary and/or sufficient conditions to be satisfied. In recent years owing to creation of computers various kinds of numerical methods have been developed which involve usually some converging processes. These methods, besides such problems as stability or error-control, can hardly give the greatest or least value, or global-optimal value for short over the whole domain, supposed to exist in advance. However, the problem becomes very agreeable if we limit ourselves to the polynomial-type case. In fact, based on the classical treatment of polynomial equations-solving in ancient China and its modernization due to J.F. Ritt, we have discovered a Finite Kernel Theorem to the effect that a finite set of real values, to be called the finite kernel set of the given problem, may be so determined that all possible extremal values will be found among this finite set and the corresponding extremal zeros are then trivially determined. Clearly it will give the global optimal value over the whole domain in consideration, if it is already known to exist in some way. Associated packages wsolve and e_val have been given by D. K. Wang and had been applied with success in various kinds of problems, polynomial-definiteness, non-linear programming, etc., particularly problems involving inequalities.