The network programming method is based on the ability to represent the objective function and the system of constraints in the form of a superposition of simpler problems. It is convenient to represent such a superposition in the form of a network, the final vertices of which correspond to variables, and the remaining vertices correspond to simpler problems included in the superposition. The solution to the problem at the input vertex gives either an estimate (upper or lower) for the original problem, or an exact solution if the network representation structure is a tree.The article discusses the application of the method for an integer linear programming problem. To do this, the coefficients of the objective function are divided arbitrarily into m parts (according to the number of problem constraints). We get m tasks (each with one constraint). These tasks are called evaluative tasks. According to the main theorem of the network programming method, the sum of the values of the objective functions in the optimal solutions of estimation problems gives the upper estimate for the solution of maximum problems (or lower for the solution of minimum problems) for the original problem. The problem of choosing a partition of the coefficients of the objective function of the original problem so that the upper estimate is minimal is called the generalized dual problem (GDV). Note that the Lagrange minimization problem in the Lagrange multiplier method is a special case of the GDV. I.V. Burkova proved that GDV is a convex programming problem.The article proposes a method for solving the GDV based on the sequential improvement of the initial solution. At each step of the method, the system of linear inequalities is solved.