Algorithm of solving the problems of the binary square programming by the method of penalty functions

In this paper the author analyses generalization of the classical quadratic programming problem, when along with linear constraints quadratic constraints are allowed. The author consider the case when variables can take only binary values 0 or 1. Such problems are important when choice of the optimal structures of systems covers a large number of variants, and the choice or refusal to select individual variants is equivalent to values 1 or 0 of the corresponding binary variables. There is described the procedure of reducing the indicated problem to the fourth-degree polynomial programming problem on the basis of the penalty method, when all terms are no greater than in the second power, except one having a fourth power. The solution of the obtained optimization problem is reduced to solving a system of equations containing only binary variables. The author proposes a heuristic recursive algorithm for solving the resulting system of equations, and variants of constructing the initial version of the solution. The parameters used in the proposed method are described. In the process of reduction, there are used the classical Cardano formulas for the roots of a cubic equation, for the practical finding of which the relations are obtained and the calculation procedure is described. The algorithm given in this paper can also be used to solve many problems of discrete mathematics.