Abstract

We now return to the general theory of extremal problems, and try to determine whether the necessary conditions for a minimum developed in Lecture 6 are also sufficient. Of course, elementary examples show that in general this is not true. Nevertheless, we shall prove that, under certain additional assumptions, the necessary extremum conditions are also sufficient, in an important class of extremal problems — convex problems. Sufficient conditions for non-convex problems can probably be formulated in terms of the second variation. However, results in this field are as yet quite sporadic, and we shall not dwell on them here.

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