Ivar Ekeland

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Research Ekeland has contributed to mathematical analysis, particularly to variational calculus and mathematical optimization.

Variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists a nearly optimal solution to a class of optimization problems.

Ekeland's variational principle can be used when the lower level set of a minimization problem is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space.

Ekeland was associated with the University of Paris when he proposed this theorem. An application of the Shapley–Folkman lemma represents the given optimal-point as a sum of points in the graphs of the original summands and of a small number of convexified summands.

This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimization methods on problems that were known to be non-convex. Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions. The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.