Optimization Methods I. Basic Concepts 1. Review of Methods to be Discussed and their Interrelations 2. A General Outline of the Calculus of Variations II. Indirect Methods 1. The Pontryagin Maximum Principle 2. Adjustment of the Calculus of Variations to the Recent Problem Formulations 3. Numerical Solution of the Boundary Value Problem for Systems of Ordinary Non Linear Differential Equations III. Direct Methods 1. Gradient Method of the First Order 2. Generalizations of the Gradient Method of the First Order and Related Methods 3. The Bellman Dynamic Programming Method Problems Comments. Variational problems which are interesting from physical and technical viewpoints are often supplemented with ordinary differential equations as constraints e. g. in the form of Newtons equations of motion. Since analytical solutions for such problems are possible only in exceptional cases and numerical treat­ ment of extensive systems of differential equations formerly caused computational difficulties in the classical calculus of variations these problems have generally been considered only with respect to their theoretical aspects. However the advent of digital computer installations has enabled us approximately since 1950 to make more practical use of the formulas provided by the calculus of variations and also to proceed from relationships which are oriented more numerically than analytically. This has proved very fruitful since there are areas in particular in automatic control and space flight technology where occasionally even relatively small optimization gains are of interest. Further on if in a problem we have a free function of time which we may choose as advantageously as possible then determination of the absolutely optimal course of this function appears always advisable even if it gives only small improve­ ments or if it leads to technical difficulties since: i) we must in any case choose some course for free functions; a criterion which gives an optimal course for that is very practical ii) also when choosing a certain technically advantageous course we mostly want to know to which extent the performance of the system can further be increased by variation of the free function.