The course of lectures describes the elements of functional and convex analysis, problems without restrictions, convex problems, smooth problems with restrictions of the type of equality and inequality for numerical functions of variables and in normalized spaces. The problems of linear programming, their solution according to the simplex method. are considered. The concept of duality is introduced, a number of methods are given for finding the initial extreme point. The acquired skills are applied to some of the most well-known types of linear programming taskstransport and assignment tasks. The following sections of the theory of extreme problems are also considered: classical variational calculus, optimal control, necessary and sufficient conditions of the extremum in classical variational calculus. Examples are given to illustrate the solution of the described types of tasks. To help teachers conducting seminars, after the theoretical part, tasks for solving at seminars, control and homework are given .