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Ivar I. Ekeland (born 2 July 1944, Paris) is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and other languages. Ekeland is known as the author of Ekeland's variational principle and for his use of the Shapley–Folkman lemma in optimization theory. He has contributed to the periodic solutions of Hamiltonian systems and particularly to the theory of Mark Krein for linear systems (Floquet theory).According to D. Pascali, writing for Mathematical Reviews ()
Ekeland is cited in the credits of Steven Spielberg's 1993 movie Jurassic Park as an inspiration of the fictional chaos theory specialist Ian Malcolm appearing in Michael Crichton's 1990 novel Jurassic Park.
Ekeland is a recipient of the D'Alembert Prize and the Jean Rostand prize. He is also a member of the Norwegian Academy of Science and Letters.
Through these writings, Ekeland had an influence on Jurassic Park, on both the novel and film. Ekeland's Mathematics and the unexpected and James Gleick's inspired the discussions of chaos theory in the novel Jurassic Park by Michael Crichton.In his afterword to Jurassic Park, acknowledges the writings of Ekeland (and James Gleick). Inside the novel, are discussed on two pages, , and chaos theory on eleven pages, including pages 75, 158, and 245:
When the novel was adapted for the film Jurassic Park by Steven Spielberg, Ekeland and Gleick were consulted by the actor Jeff Goldblum as he prepared to play the mathematician specializing in chaos theory.:
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's principle leads to a quick proof of the Caristi fixed point theorem. (1990). 9780521382892, Cambridge University Press. ISBN 9780521382892
Ekeland was associated with the University of Paris when he proposed this theorem.
For example, problems of linear optimization are separable. For a separable problem, we consider an optimal solution
with the minimum value For a separable problem, we consider an optimal solution ( xmin, f( xmin)) to the " convexified problem", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the limit of a sequence of points in the convexified problem
The limit of a sequence is a member of the closure of the original set, which is the smallest closed set that contains the original set. The Minkowski sum of two need not be closed, so the following inclusion can be strict
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 iterative method on problems that were known to be non-convex.: . Lemaréchal's experiments were discussed in later publications:
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: (1993). 9783540568520, Springer-Verlag. ISBN 9783540568520
: Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging Lemaréchal's experiments on page 373. Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions.:
and also considered the convex closure of a problem of non-convex minimization—that is, the problem defined by the closed convex hull closure operator of the epigraph of the original problem. Their study of duality gaps was extended by Di Guglielmo to the quasiconvex closure of a non-convex minimization problem—that is, the problem defined by the closed convex hullclosure operator of the lower level set:
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describes an application of [[Lagrangian dual|dual problem]] methods to the scheduling of electrical power plants ("unit commitment problems"), where non-convexity appears because of integer constraints:
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