The paper considers the quadratic programming problem with a strictly convex separable objective function, a single linear constraint, and two-sided constraints on variables. This problem is commonly called the convex separable quadratic knapsack problem (CQKnP). We are interested in an algorithm for solving the CQKnP with linear complexity. The works devoted to this topic contain inaccuracies in the description of algorithms and inefficient implementations. In this work, the existing difficulties are overcome.

One of the main tasks of mathematical diagnostics is the strict separation of two finite sets in a Euclidean space. Strict linear separation is widely known and reduced to the solution of a linear programming problem. We introduce the notion of strict polynomial separation and show that the strict polynomial separation of two sets can be also reduced to the solution of a linear programming problem. The objective function of the linear programming problem proposed in this paper has the following feature: its optimal value can be only zero or one, i.e., it is zero if the sets admit strict polynomial separation and one otherwise. Some illustrative examples of the strict separation of two sets on a plane with the use of fourth degree algebraic polynomials in two variables are given. The application efficiency of strict polynomial separation to binary data classification problems is analyzed.

A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen's inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.