Strict polynomial separation of two sets

Abstract

One of the main tasks of mathematical diagnostics is the strict separation of two finite sets in Euclidean space. A strict linear separation is widely known and is reduced to a linear programming problem. We introduce a strict polynomial separation and show that the strict polynomial separation of two sets also is reduced to a linear programming problem. The objective function of the linear programming problem proposed in this paper has the following feature: its optimal value can only be zero or one. It is zero if the sets are strictly polynomially separable; otherwise, it is one. We give illustrative examples of strict separation of two sets with the help of two-variable fourth-degree algebraic polynomials on the plane. We also analyze the efficiency of applying the strict polynomial separation to binary classification problems.