Abstract:

In many situations, the best solutions of some problems are to be found. The problems are usually replaced by their mathematical models and become mathematical problems. The branch of mathematics studying this topic is theory of optimization. In this contribution we concentrate on the investigation of problems modelled by real functions of several variables.
In this case, finding the best solution means to determine global extrema of an objective function. The situation simplifies substantially if it is known that global extrema exist. Then it is sufficient to find critical points, i.e. points in which local extrema can occur, to decide whether they are points of local extrema and choose those giving the greatest or smallest value. The standard result guaranteeing the existence of global extrema is the well-known Weierstrass Theorem: Any function continuous on a compact domain assumes its greatest and smallest value.
The situation complicates when the existence of global extrema is not guaranteed. There are no universal methods how to proceed in such cases. In the paper, a few possible approaches are presented and their use on examples is demonstrated. They include namely solutions based on monotonicity and some important inequalities. Likewise their efficiency and the comparison with standard tools of differential calculus for functions of several variables for finding local extrema (first and second order conditions) are discussed.

Keywords:

global extrema, Weierstrass Theorem, arithmetical and geometrical inequality