AbstractThe Tent Method solves different extremal problems (minima and maxima of smooth functions, minimax problems, optimization problems, etc.). A detailed description of the Method with its geometrical and topological foundations is given. The Tent Method has application to optimization problems; in particular, it allows to prove the Maximum Principle. The last section contains the sufficient condition of optimality (in a revised form) and the robust optimization problems. We show that the regular synthesis (which is the main instrument in the sufficient conditions of optimality) ensures the robustness. The text is illustrated by figures and examples. Copyright © 2004 John Wiley & Sons, Ltd.