Ivar Ekeland (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Ivar Ekeland" in English language version.

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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.

ams.org

mathscinet.ams.org

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. 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.
According to Jeremy Gray, writing for Mathematical Reviews (MR945956)
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.
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.
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.
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.
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.
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 & 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 & Temam (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 convex hull 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.
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 (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 10.1.1.158.1736. doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.

archive.org

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. 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.
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.

books.google.com

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.
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.

dnva.no

"Group 1: Mathematical studies". Norwegian Academy of Science and Letters. Archived from the original on 27 September 2011. Retrieved 12 April 2011.

doi.org

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 & 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 & Temam (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 convex hull 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.
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 (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 10.1.1.158.1736. doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.

jstor.org

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 & Temam (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 convex hull 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.

mathprog.org

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.

mit.edu

web.mit.edu

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 (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 10.1.1.158.1736. doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.

nyu.edu

homepages.nyu.edu

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.

psu.edu

citeseerx.ist.psu.edu

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 (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 10.1.1.158.1736. doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.

semanticscholar.org

api.semanticscholar.org

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 (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 10.1.1.158.1736. doi:10.1109/tac.1983.1103136. S2CID 6329622. Retrieved 2 February 2011.

web.archive.org

"Group 1: Mathematical studies". Norwegian Academy of Science and Letters. Archived from the original on 27 September 2011. Retrieved 12 April 2011.

worldcat.org

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 (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.