Popular science: Jurassic Park by Crichton and Spielberg
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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]}
In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland,^{[9]}^{[10]}^{[11]} is a theorem that asserts that there exists a nearly optimal solution to a class of optimization problems.^{[12]}
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
$x_{\min }=(x_{1},\dots ,x_{N})_{\min }$
with the minimum value f(x_{min}). For a separable problem, we consider an optimal solution (x_{min}, f(x_{min}))
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
$(x_{j},f(x_{j}))\in \mathrm {Conv} (\mathrm {Graph} (f_{n})).\,$^{[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 minimizationmethods 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]}
Bibliography
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Research
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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.)
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.
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.
Aubin, Jean-Pierre; Ekeland, Ivar (2006). Applied nonlinear analysis. Mineola, NY: Dover Publications, Inc. pp. x+518. ISBN 978-0-486-45324-8. MR 2303896. (Reprint of the 1984 Wiley (MR749753) ed.)
Exposition for a popular audience
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Ekeland, Ivar (1988). Mathematics and the unexpected (Translated by Ekeland from his French ed.). Chicago, IL: University Of Chicago Press. pp. xiv+146. ISBN 978-0-226-19989-4. MR 0945956.
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.
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.
^ ^{a}^{b}^{c}^{d}Ekeland (1988, Appendix 2 The Feigenbaum bifurcation, pp. 132–138) describes the chaotic behavior of the iteratedlogistic 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.
^ ^{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.
^ ^{a}^{b}Jones (1993, p. 9): Jones, Alan (August 1993). Clarke, Frederick S. (ed.). "Jurassic Park: Computer graphic dinosaurs". Cinefantastique. 24 (2). Frederick S. Clarke: 8–15. ASIN B002FZISIO. Retrieved 2011-04-12.
^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.
^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.
^ ^{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.
^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.
^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.
^ ^{a}^{b}Kirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 978-0-521-38289-2.
^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.
^ ^{a}^{b}^{c}^{d}(Ekeland & Temam 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.
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. Rockafellar, R. Tyrrell (1997) [1970]. Convex analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.
^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.
^ ^{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.
^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.
^Bertsekas (1996, pp. 364–381)acknowledging Ekeland & Temam (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, 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. CiteSeerX10.1.1.158.1736. doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.
^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.