Ivar Ekeland

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Picture of the Julia set
Ivar Ekeland has written popular books about chaos theory and about fractals,[1][2] such as the Julia set (animated). Ekeland's exposition provided mathematical inspiration to Michael Crichton's discussion of chaos in Jurassic Park.[3]

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 Kreĭn indices for linear systems (Floquet theory).[4] Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.[3]


Ekeland studied at the École Normale Supérieure (1963–1967). He is a senior research fellow at the French National Centre for Scientific Research (CNRS). He obtained his doctorate in 1970. He teaches mathematics and economics at the Paris Dauphine University, the École Polytechnique, the École Spéciale Militaire de Saint-Cyr, and the University of British Columbia in Vancouver. He was the chairman of Paris-Dauphine University from 1989 to 1994.

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.[5]

Popular science: Jurassic Park by Crichton and Spielberg[edit]

Picture of Jeff Goldblum
Actor Jeff Goldblum consulted Ekeland while preparing to play a mathematician specializing in chaos theory in Spielberg's Jurassic Park.[6]

Ekeland has written several books on popular science, in which he has explained parts of dynamical systems, chaos theory, and probability theory.[1][7][8] These books were first written in French and then translated into English and other languages, where they received praise for their mathematical accuracy as well as their value as literature and as entertainment.[1]

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 Chaos inspired the discussions of chaos theory in the novel Jurassic Park by Michael Crichton.[3] 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.[6]


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

Variational principle[edit]

In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland,[9][10][11] is a theorem that asserts that there exist a nearly optimal solution to a class of optimization problems.[12]

Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space.[13]

Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.[13][14]

Ekeland was associated with the University of Paris when he proposed this theorem.[9]

Variational theory of Hamiltonian systems[edit]

Ivar Ekeland is an expert on variational analysis, which studies mathematical optimization of spaces of functions. His research on periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems (Floquet theory) was described in his monograph.[4]

Additive optimization problems[edit]

The Shapley–Folkman lemma depicted by a diagram with two panes, one on the left and the other on the right. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. Comparing the left array and the right pane, one confirms that the right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.
Ivar Ekeland applied the Shapley–Folkman lemma to explain Claude Lemarechal's success with Lagrangian relaxation on non-convex minimization problems. This lemma concerns the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowski sum of the four non-convex sets (right) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown in red).

Ekeland explained the success of methods of convex minimization on large problems that appeared to be non-convex. In many optimization problems, the objective function f are separable, that is, the sum of many summand-functions each with its own argument:

For example, problems of linear optimization are separable. For a separable problem, we consider an optimal solution

with the minimum value f(xmin). For a separable problem, we consider an optimal solution (xminf(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

[15][16] 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.[17][15][18] Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions.[15][18][19] The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.[15][20][21][22]



  • Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. Vol. 28. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). ISBN 978-0-89871-450-0. MR 1727362. (Corrected reprinting of the 1976 North-Holland (MR463993) ed.)
The book is cited over 500 times in MathSciNet.

Exposition for a popular audience[edit]

Picture of the Feigenbaum bifurcation of the iterated logistic-function
The Feigenbaum bifurcation of the iterated logistic function system was described as an example of chaos theory in Ekeland's Mathematics and the unexpected.[1]

See also[edit]


  1. ^ a b c d Ekeland (1988, Appendix 2 The Feigenbaum bifurcation, pp. 132–138) describes the chaotic behavior of the iterated logistic function, which exhibits the Feigenbaum bifurcation. A paperback edition was published: Ekeland, Ivar (1990). Mathematics and the unexpected (Paperback ed.). University Of Chicago Press. ISBN 978-0-226-19990-0.
  2. ^ According to Jeremy Gray, writing for Mathematical Reviews (MR945956)
  3. ^ a b c In his afterword to Jurassic Park, Crichton (1997, pp. 400) acknowledges the writings of Ekeland (and Gleick). Inside the novel, fractals are discussed on two pages, (Crichton 1997, pp. 170–171), and chaos theory on eleven pages, including pages 75, 158, and 245:
    Crichton, Michael (1997). Jurassic Park. Ballantine Books. ISBN 9780345418951. Retrieved 2011-04-19.
  4. ^ a b According to D. Pascali, writing for Mathematical Reviews (MR1051888)
    Ekeland, Ivar (1990). Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 19. Berlin: Springer-Verlag. pp. x+247. ISBN 978-3-540-50613-3. MR 1051888.
  5. ^ "Group 1: Mathematical studies". Norwegian Academy of Science and Letters. Archived from the original on 27 September 2011. Retrieved 12 April 2011.
  6. ^ a b Jones (1993, p. 9): Jones, Alan (August 1993). Clarke, Frederick S. (ed.). "Jurassic Park: Computer graphic dinosaurs". Cinefantastique. Frederick S. Clarke. 24 (2): 8–15. ASIN B002FZISIO. Retrieved 2011-04-12.
  7. ^ According to Mathematical Reviews (MR1243636) discussing Ekeland, Ivar (1993). The broken dice, and other mathematical tales of chance (Translated by Carol Volk from the 1991 French ed.). Chicago, IL: University of Chicago Press. pp. iv+183. ISBN 978-0-226-19991-7. MR 1243636.
  8. ^ According to Mathematical Reviews (MR2259005) discussing Ekeland, Ivar (2006). The best of all possible worlds: Mathematics and destiny (Translated from the 2000 French ed.). Chicago, IL: University of Chicago Press. pp. iv+207. ISBN 978-0-226-19994-8. MR 2259005.
  9. ^ a b Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47 (2): 324–353. doi:10.1016/0022-247X(74)90025-0. ISSN 0022-247X.
  10. ^ Ekeland, Ivar (1979). "Nonconvex minimization problems". Bulletin of the American Mathematical Society. New Series. 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. MR 0526967.
  11. ^ Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. Vol. 28 (Corrected reprinting of the (1976) North-Holland ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 978-0-89871-450-0. MR 1727362.
  12. ^ Aubin, Jean-Pierre; Ekeland, Ivar (2006). Applied nonlinear analysis (Reprint of the 1984 Wiley ed.). Mineola, NY: Dover Publications, Inc. pp. x+518. ISBN 978-0-486-45324-8. MR 2303896.
  13. ^ a b Kirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 978-0-521-38289-2.
  14. ^ Ok, Efe (2007). "D: Continuity I" (PDF). Real Analysis with Economic Applications. Princeton University Press. p. 664. ISBN 978-0-691-11768-3. Retrieved January 31, 2009.
  15. ^ a b c d (Ekeland 1999, pp. 357–359): Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging Lemaréchal's experiments on page 373.
  16. ^ 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 closed sets need not be closed, so the following inclusion can be strict
    Clos(P) + Clos(Q) ⊆ Clos( Clos(P) + Clos(Q) );
    the inclusion can be strict even for two convex closed summand-sets, according to Rockafellar (1997, pp. 49 and 75). Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergent sequences.
  17. ^ Lemaréchal (1973, p. 38): Lemaréchal, Claude (April 1973), Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems] (in French), Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France: IRIA (now INRIA), Laboratoire de recherche en informatique et automatique, p. 41{{citation}}: CS1 maint: location (link). Lemaréchal's experiments were discussed in later publications:
    Aardal (1995, pp. 2–3): Aardal, Karen (March 1995). "Optima interview Claude Lemaréchal" (PDF). Optima: Mathematical Programming Society Newsletter. 45: 2–4. Retrieved 2 February 2011.

    Hiriart-Urruty & Lemaréchal (1993, pp. 143–145, 151, 153, and 156): Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1993). "XII Abstract duality for practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 306. Berlin: Springer-Verlag. pp. 136–193 (and bibliographical comments on pp. 334–335). ISBN 978-3-540-56852-0. MR 1295240.
  18. ^ a b Ekeland, Ivar (1974). "Une estimationa priori en programmation non convexe". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B (in French). 279: 149–151. ISSN 0151-0509. MR 0395844.
  19. ^ Aubin & Ekeland (1976, pp. 226, 233, 235, 238, and 241): Aubin, J. P.; Ekeland, I. (1976). "Estimates of the duality gap in nonconvex optimization". Mathematics of Operations Research. 1 (3): 225–245. doi:10.1287/moor.1.3.225. JSTOR 3689565. MR 0449695.
    Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) also considered the convex closure of a problem of non-convex minimization—that is, the problem defined by the closed convex hull 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 convexhull of the lower level sets:

    Di Guglielmo (1977, pp. 287–288): Di Guglielmo, F. (1977). "Nonconvex duality in multiobjective optimization". Mathematics of Operations Research. 2 (3): 285–291. doi:10.1287/moor.2.3.285. JSTOR 3689518. MR 0484418.
  20. ^ Aubin (2007, pp. 458–476): Aubin, Jean-Pierre (2007). "14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463-465)". Mathematical methods of game and economic theory (Reprint with new preface of 1982 North-Holland revised English ed.). Mineola, NY: Dover Publications, Inc. pp. xxxii+616. ISBN 978-0-486-46265-3. MR 2449499.
  21. ^ Bertsekas (1996, pp. 364–381)acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
    Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods (Reprint of (1982) Academic Press ed.). Belmont, MA: Athena Scientific. pp. xiii+395. ISBN 978-1-886529-04-5. MR 0690767.

    Bertsekas (1996, pp. 364–381) describes an application of Lagrangian dual methods to the scheduling of electrical power plants ("unit commitment problems"), where non-convexity appears because of integer constraints:

    Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R. Jr.; Posbergh, Thomas A. (January 1983). "Optimal short-term scheduling of large-scale power systems" (PDF). IEEE Transactions on Automatic Control. AC-28 (1): 1–11. CiteSeerX doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.
  22. ^ Bertsekas (1999, p. 496): Bertsekas, Dimitri P. (1999). "5.1.6 Separable problems and their geometry". Nonlinear Programming (Second ed.). Cambridge, MA.: Athena Scientific. pp. 494–498. ISBN 978-1-886529-00-7.

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