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词条 Ivar Ekeland
释义

  1. Biography

  2. Popular science: Jurassic Park by Crichton and Spielberg

  3. Research

     Variational principle  Variational theory of Hamiltonian systems  Additive optimization problems 

  4. Bibliography

     Research  Exposition for a popular audience 

  5. See also

  6. Notes

  7. External links

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).[3] Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.[3]

Biography

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.[4]

Popular science: Jurassic Park by Crichton and Spielberg

Ekeland has written several books on popular science, in which he has explained parts of dynamical systems, chaos theory, and probability theory.[1][5][6] 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.[7] 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.[8]

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,[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.[18]

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

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.[3]

Additive optimization problems

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 {{nowrap|f(xmin).}} 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

[23][15] 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.[16][17][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.[17][18][19] The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.[17][20][21][22]

Bibliography

Research

  • {{cite book|last1=Ekeland|first1=Ivar|last2=Temam|first2=Roger|authorlink2=Roger Temam|title=Convex analysis and variational problems|edition=Corrected reprinting of the 1976 North-Holland ({{MR|463993}})|series=Classics in applied mathematics|volume=28 |publisher=Society for Industrial and Applied Mathematics (SIAM)|location=Philadelphia, PA|year=1999|isbn=978-0-89871-450-0|mr=1727362|ref=harv}}

The book is cited over 500 times in MathSciNet.

  • {{cite journal|last=Ekeland|first=Ivar|title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|ref=harv}}
  • {{cite book|last=Ekeland|first=Ivar|title=Convexity methods in Hamiltonian mechanics|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]|volume=19|publisher=Springer-Verlag|location=Berlin|year=1990|pages=x+247|isbn=978-3-540-50613-3|mr=1051888|ref=harv}}
  • {{cite book|last1=Aubin|first1=Jean-Pierre|last2=Ekeland|first2=Ivar|title=Applied nonlinear analysis|edition=Reprint of the 1984 Wiley ({{MR|749753}})|publisher=Dover Publications, Inc.|location=Mineola, NY|year=2006|pages=x+518|isbn=978-0-486-45324-8|mr=2303896|ref=harv}}

Exposition for a popular audience

  • {{cite book|title=Mathematics and the unexpected|first=Ivar|last=Ekeland||pages=xiv+146|publisher=University Of Chicago Press|location=Chicago, IL|year=1988|isbn=978-0-226-19989-4|ref=harv|mr=945956|edition=Translated by Ekeland from his French}}
  • {{cite book| last=Ekeland|first=Ivar |title=The broken dice, and other mathematical tales of chance||mr=1243636|edition=Translated by Carol Volk from the 1991 French|publisher=University of Chicago Press|location=Chicago, IL|year=1993|pages=iv+183|isbn=978-0-226-19991-7|ref=harv}}
  • {{cite book|last=Ekeland|first=Ivar |title=The best of all possible worlds: Mathematics and destiny|edition=Translated from the 2000 French|publisher=University of Chicago Press|location=Chicago, IL|year=2006|pages=iv+207|isbn=978-0-226-19994-8|mr=2259005|ref=harv}}

See also

  • Jonathan M. Borwein ("smooth" variational principle)
  • Robert R. Phelps (a "grandfather" of variational principles)
  • David Preiss ("smooth" variational principle)
{{clear}}

Notes

1. ^1 2 3 {{harvtxt|Ekeland|1988|loc=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: {{cite book|first=Ivar|last=Ekeland||title= Mathematics and the unexpected|edition=Paperback|publisher=University Of Chicago Press|year=1990|isbn=978-0-226-19990-0|ref=harv}}
2. ^According to Jeremy Gray, writing for Mathematical Reviews ({{MR|945956}})
3. ^1 According to D. Pascali, writing for Mathematical Reviews ({{MR|1051888}})
{{harvtxt|Ekeland|1990}} {{cite book|last=Ekeland|first=Ivar|title=Convexity methods in Hamiltonian mechanics|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]|volume=19|publisher=Springer-Verlag|location=Berlin|year=1990|pages=x+247|isbn=978-3-540-50613-3|mr=1051888|ref=harv}}
4. ^{{cite web|title=Group 1: Mathematical studies|url=http://www.dnva.no/c40134/artikkel/vis.html?tid=40147|publisher=Norwegian Academy of Science and Letters|accessdate=12 April 2011}}
5. ^According to Mathematical Reviews ({{MR|1243636}}) discussing {{cite book|last=Ekeland|first=Ivar|title=The broken dice, and other mathematical tales of chance|mr=1243636|edition=Translated by Carol Volk from the 1991 French|publisher=University of Chicago Press|location=Chicago, IL|year=1993|pages=iv+183|isbn=978-0-226-19991-7|ref=harv}}
6. ^According to Mathematical Reviews ({{MR|2259005}}) discussing {{cite book|last=Ekeland|first=Ivar|title=The best of all possible worlds: Mathematics and destiny|edition=Translated from the 2000 French|publisher=University of Chicago Press|location=Chicago, IL|year=2006|pages=iv+207|isbn=978-0-226-19994-8|mr=2259005|ref=harv}}
7. ^1 2 In his afterword to Jurassic Park, {{harvtxt|Crichton|1997|pp=400}} acknowledges the writings of Ekeland (and Gleick). Inside the novel, fractals are discussed on two pages, {{harv|Crichton|1997|pp=170–171}}, and chaos theory on [https://books.google.com/books?ei=URukTaaYF9rO4wbe2v2nCg&ct=result&id=O8XZAAAAMAAJ&dq=Ivar+Ekeland%2C+Jurassic+Park&q=chaos#search_anchor eleven pages, including pages 75, 158, and 245]:
{{cite book|first=Michael|last=Crichton|authorlink=Michael Chriton|title=Jurassic Park|year=1997|edition=|publisher=Ballantine Books|url=https://books.google.com/?id=O8XZAAAAMAAJ&dq=Ivar+Ekeland%2C+Jurassic+Park&q=fractal+OR+Ekeland#search_anchor|accessdate=2011-04-19|ref=harv|isbn=9780345418951}}
8. ^1 {{harvtxt|Jones|1993|p=9}}: {{cite journal|last=Jones|first=Alan|journal=Cinefantastique|volume=24|number=2|pages=8–15|date=August 1993|title=Jurassic Park: Computer graphic dinosaurs |url=https://books.google.com/?id=yS4nAQAAIAAJ&dq=Ivar+Ekeland%2C+Jurassic+Park&q=Ivar+Ekeland#search_anchor |asin=B002FZISIO|publisher=Frederick S. Clarke|editor-first=Frederick S.|editor-last=Clarke|accessdate=2011-04-12|ref=harv|}}
9. ^1 {{cite journal | doi = 10.1016/0022-247X(74)90025-0 | last = Ekeland | first = Ivar | title = On the variational principle | journal = J. Math. Anal. Appl. | volume = 47 | issue = 2 | year = 1974 | pages = 324–353 | issn = 0022-247X}}
10. ^{{cite journal|last=Ekeland|first=Ivar|title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|ref=harv}}
11. ^{{cite book|last1=Ekeland|first1=Ivar|last2=Temam|first2=Roger|authorlink=Roger Temam|title=Convex analysis and variational problems|edition=Corrected reprinting of the (1976) North-Holland|series=Classics in applied mathematics|volume=28 |publisher=Society for Industrial and Applied Mathematics (SIAM)|location=Philadelphia, PA|year=1999|pages=357–373|isbn=978-0-89871-450-0|mr=1727362|ref=harv}}
12. ^{{cite book|last1=Aubin|first1=Jean-Pierre|last2=Ekeland|first2=Ivar|title=Applied nonlinear analysis|edition=Reprint of the 1984 Wiley|publisher=Dover Publications, Inc.|location=Mineola, NY|year=2006|pages=x+518|isbn=978-0-486-45324-8|mr=2303896|ref=harv}}
13. ^1 {{cite book |author1=Kirk, William A. |author2=Goebel, Kazimierz | title = Topics in Metric Fixed Point Theory | year = 1990 | publisher = Cambridge University Press | isbn = 978-0-521-38289-2}}
14. ^{{cite book|last=Ok|first=Efe|title=Real Analysis with Economic Applications|publisher=Princeton University Press|year=2007|pages=664|chapter=D: Continuity I|isbn=978-0-691-11768-3|chapter-url=http://homepages.nyu.edu/~eo1/Book-PDF/Ekeland.pdf|accessdate=January 31, 2009}}
15. ^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 {{harvtxt|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.
16. ^{{harvtxt|Lemaréchal|1973|p=38}}: {{citation|last=Lemaréchal|first=Claude|authorlink=Claude Lemaréchal|title=Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems]|language=French| date=April 1973 |number=16|location=Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France|publisher=IRIA (now INRIA), Laboratoire de recherche en informatique et automatique|pages=41|ref=harv}}. Lemaréchal's experiments were discussed in later publications:
{{harvtxt|Aardal|1995|pp=2–3}}: {{cite journal|first=Karen|last=Aardal|authorlink=Karen Aardal|title=Optima interview Claude Lemaréchal|journal=Optima: Mathematical Programming Society Newsletter|pages=2–4|date=March 1995|volume=45|url=http://www.mathprog.org/Old-Optima-Issues/optima45.pdf|accessdate=2 February 2011|ref=harv}}

{{harvtxt|Hiriart-Urruty|Lemaréchal|1993|pp=143–145, 151, 153, and 156}}: {{cite book|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|authorlink2=Claude Lemaréchal|chapter=XII Abstract duality for practitioners|title=Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods|series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]|volume=306|publisher=Springer-Verlag|location=Berlin|year=1993|pages=136–193 (and bibliographical comments on pp. 334–335)|isbn=978-3-540-56852-0|mr=1295240}}
17. ^1 2 3 {{harv|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.
18. ^1 {{cite journal|last=Ekeland|first=Ivar|title=Une estimationa priori en programmation non convexe|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences|series=Séries A et B|language=French|issn=0151-0509|volume=279|year=1974|pages=149–151|mr=395844|ref=harv}}
19. ^{{harvtxt|Aubin|Ekeland|1976|pp=226, 233, 235, 238, and 241}}: {{cite journal|last1=Aubin|first1=J. P.|last2=Ekeland|first2=I.|issue=3|journal=Mathematics of Operations Research|pages=225–245|title=Estimates of the duality gap in nonconvex optimization|volume=1| year = 1976|doi=10.1287/moor.1.3.225|mr=449695|jstor=3689565|ref=harv}}
{{harvtxt|Aubin|Ekeland|1976}} and {{harvtxt|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:

{{harvtxt|Di Guglielmo|1977|pp=287–288}}: {{cite journal|last=Di Guglielmo|first=F.|title=Nonconvex duality in multiobjective optimization|doi=10.1287/moor.2.3.285|volume=2|year=1977|number=3|pages=285–291|journal=Mathematics of Operations Research|mr=484418|jstor=3689518}}
20. ^{{harvtxt|Aubin|2007|pp=458–476}}: {{cite book|last=Aubin|first=Jean-Pierre|chapter=14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463-465)|title=Mathematical methods of game and economic theory|edition=Reprint with new preface of 1982 North-Holland revised English|publisher=Dover Publications, Inc|location=Mineola, NY|year=2007|pages=xxxii+616|isbn=978-0-486-46265-3|mr=2449499|ref=harv}}
21. ^{{harvtxt|Bertsekas|1996|pp=364–381}}acknowledging {{harvtxt|Ekeland|1999}} on page 374 and {{harvtxt|Aubin|Ekeland|1976}} on page 381:
{{cite book|last=Bertsekas|first=Dimitri P.|authorlink=Dimitri P. Bertsekas|chapter=5.6 Large scale separable integer programming problems and the exponential method of multipliers|title=Constrained optimization and Lagrange multiplier methods|edition=Reprint of (1982) Academic Press|year=1996|location=Belmont, MA|isbn=978-1-886529-04-5|pages=xiii+395|publisher=Athena Scientific|mr=690767|ref=harv}}

{{harvtxt|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:

{{cite journal|journal=IEEE Transactions on Automatic Control|volume=AC-28|number=1|date=January 1983|title=Optimal short-term scheduling of large-scale power systems|first1=Dimitri P.|last1=Bertsekas|authorlink1=Dimitri Bertsekas|first2=Gregory S.|last2=Lauer|first3=Nils R., Jr.|last3=Sandell|first4=Thomas A.|last4=Posbergh|pages=1–11|ref=harv|url=http://web.mit.edu/dimitrib/www/Unit_Comm.pdf |accessdate=2 February 2011|doi=10.1109/tac.1983.1103136|citeseerx=10.1.1.158.1736}}
22. ^{{harvtxt|Bertsekas|1999|p=496}}: {{cite book|last=Bertsekas|first=Dimitri P.|authorlink=Dimitri P. Bertsekas |title=Nonlinear Programming|edition=Second|chapter=5.1.6 Separable problems and their geometry|pages=494–498|publisher=Athena Scientific|year=1999|location=Cambridge, MA.|isbn =978-1-886529-00-7}}

External links

{{wikiquote}}
  • {{MathGenealogy|id=59369}}
  • Ekeland's webpage at CEREMADE
  • Conference on "Economics and Mathematics" by Ivar Ekeland , held at Canal U (2000)
  • [https://web.archive.org/web/20110926222914/http://www.iecn.u-nancy.fr/~hijazi/CVEKELAND.html Ekeland's Curriculum vitae]
{{Authority control}}{{DEFAULTSORT:Ekeland, Ivar}}

15 : Variational analysts|Functional analysts|Mathematical economists|20th-century French mathematicians|21st-century French mathematicians|Canadian mathematicians|French people of Norwegian descent|Canadian people of Norwegian descent|Canadian people of French descent|1944 births|Living people|École Normale Supérieure alumni|University of British Columbia faculty|Canada Research Chairs|Textbook writers
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