The manual discusses the most fundamental results of mathematical programming, classical variational calculus and optimal control, which form the main course of optimization methods. The reasons for convex analysis, problems of extension of extreme problems and the theory of the existence of solutions, sufficient conditions of the extremum, the concept of dynamic programming and numerical methods of solution. are presented. A number of extreme problems arising in the theory of space science have been solved, some classical inequalities have been proved by methods of the theory of extreme problems. The book includes more than 400 tasks, which allows you to use the manual in practical classes and workshops .