Application of topology in optimization theory

In 1958 the author gave a proof of the Maximum Principle [Dokl. Akad. Nauk SSSR 119 (6) (1958) 1070–1073] in the Theory of Optimal Control. B. Pshenichni had written that the proof was a little sensational, since it used topology to obtain a result of variational calculus. Later the author worked out the Tent Method [Russian Math. Surveys 30 (3) (1975) 1–54] that is a general way to solve some extremal problems. In fact, main ideas of the Method were contained in the proof of the Maximum Principle [Dokl. Akad. Nauk SSSR 119 (6) (1958) 1070–1073]. In the talk a short survey of the Tent Method and the idea of the proof of the Maximum Principle are explained, paying attention to necessary topological tools.