A method of unconstrained global optimization

It is shown that a global optimization method for fourth degree polynomials can solve systems of polynomial equations in many variables of any degree.

Abstract

The method that is defined in the following finds the maximum or minimum of a real-valued function of many variables even if the function has local maxima or minima. The methods is iterative and guaranteed to converge for polynomials in several variables up to fourth degree. It can also be used successfully for other types of functions. The method approximates a function automatically if it is not a polynomial of degree four or less. It is shown that a global optimization method for fourth degree polynomials can solve systems of polynomial equations in many variables of any degree. The speed of convergence is analysed theoretically and empirically. The method has been applied by the author and collaborators to the solution of systems of nonlinear equations in many variables (up to 100 variables), determination of rate constants in nonlinear differential equations (systems identification), chemical equilibrium equations, curve fitting of sums of exponentials, pattern recognition, and analysis of spectra with nonlinear superposition (Bremermann and Lam [11]). The applications will be reported elsewhere.