Advances in Mathematical Economics Volume 20 pp 131-150

Part of the Advances in Mathematical Economics book series (MATHECON, volume 20) | Cite as

Survey of the Theory of Extremal Problems

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Abstract

In the paper some general principles of the theory of extremum are considered, and basing of these principles we give a survey of fundamental results on the foundation of the theory, conditions of extrema and existence of solutions.

Keywords

Theory of extremal problems Implicit function Convex analysis Lagrange principle Fields theory Existence of solution 

References

  1. 1.
    Euclid (2007) Euclid’s elements. Green Lion Press, Ann ArborGoogle Scholar
  2. 2.
    Alexeev V, Tikhomirov V, Fomin S (1987) Optimal control. Plenum Publishing Corporation, New YorkGoogle Scholar
  3. 3.
    Banach S (1932) “Théorie des opérations linéaire”, Warszava, Monographje MatematyczneMATHGoogle Scholar
  4. 4.
    Bernoulli I (1696) Problema novum, ad cujus solutionem Matematici invitantur. Acta EruditorumGoogle Scholar
  5. 5.
    Bliss GA (1963) Lectures on the calculus of variations. University of Chicago PressGoogle Scholar
  6. 6.
    de Fermat P (1891) Oeuvres de Fermat, vol 1. Gauthier-Villars, ParisMATHGoogle Scholar
  7. 7.
    Dini U (1877/1878) Analisi infinitesimale. Lezzione dettate nella Università Pisa. Bd 2Google Scholar
  8. 8.
    Dmitruk AV, Milyutin AA, Osmolovskii NP (1980) Lyustrenik’s theorem and the theory of extrema. Uspehi Mat Nauk 35(6): 11–46MathSciNetMATHGoogle Scholar
  9. 9.
    Dantzig GB (1963) Linear Programming and Extensions. Princeton University Press, Princeton, NJMATHGoogle Scholar
  10. 10.
    Dubovitskii AY, Milyutin AA (1965) Extremum problems with constraints. Zh Vychisl Mat i Mat Fiz 5:395–453Google Scholar
  11. 11.
    Euler L (1744) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latssimo sencu accepti. LausanneMATHGoogle Scholar
  12. 12.
    Fenchel W (1953) Convex Cones, Sets, and Functions. Princeton University Department of Mathematics, Princeton, NJGoogle Scholar
  13. 13.
    Frèchet V (1912) Sur la notion de differentielle. Nouvelle annale de mathematique, Ser. 4, V.XII. S. 845Google Scholar
  14. 14.
    Graves LM (1950) Some mapping theorems. Duke Math J 17:111–114MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hamilton WR (1835) Second essay on a general methods in dynamics. Philos Trans R Soc Pt 1:95–144CrossRefGoogle Scholar
  16. 16.
    Hilbert D. Hazewinkel M (ed) (2001) “Hilbert problems”, Encyclopedia of mathematics. Springer. ISBN:978-1-55608-010-4Google Scholar
  17. 17.
    Ioffe A, Tikhomirov V (1979) Theory of extremal problems. North-Holland, AmsterdamGoogle Scholar
  18. 18.
    Jacobi CGJ (1837) Zur Theorie der Variations-Rechnung und der Differential-Gleichungen. Krelle’s Journall 17:68–82MathSciNetGoogle Scholar
  19. 19.
    John F Extremal problems with inequalities as subsidery conditions. In: Studies and Essays. Courant Anniverrsary Volume, 1948, pp 187–204Google Scholar
  20. 20.
    Kantorovich LV (1939) Mathematical methods of organizing and planning production. Manag Sci 6(4)(Jul., 1960):366–422MathSciNetGoogle Scholar
  21. 21.
    Karush WE (1939) Minima of functions of several variables with inequalities as side conditions, University of Chicago PressGoogle Scholar
  22. 22.
    Kepler I (1615) The volume of a Wine Barrel – Kepler’s Nova stereometria doliorum vinariorum, Lincii (Roberto Cardil Matematicas Visuales)Google Scholar
  23. 23.
    Kneser A (1925) Lehrbuch der Variationsrechnung. Springer, WiesbadenCrossRefMATHGoogle Scholar
  24. 24.
    Kuhn HW, Tucker AW (1951) Nonlinear programming. University of California Press, Berkley, pp 481–482MATHGoogle Scholar
  25. 25.
    Lagrange JL (1766) Essai d’une nouvelle méthode pour determiner les maxima et les minima periales Petropolitanae, vol 10, 51–93Google Scholar
  26. 26.
    Lagrange JL (1797) Théorie des fonctions analytiques, ParisGoogle Scholar
  27. 27.
    Leach E (1961) A note on inverse mapping theorem. Proc AMS 12:694–697MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Legendre AM (1786) Mémoire sur la maniere de distingue les maxima des minima dans le calcul de variations/ Histoire de l’Academie Royallle des Sciences. Paris, pub 1788. 7–37Google Scholar
  29. 29.
    Leibniz G (1684) Acta Eroditorum, L.M.S., t. V, pp 220–226Google Scholar
  30. 30.
    Levin AY On an algorithm for the minimization of convex function Sov. Math., Dokl. – 1965. – no. 6, pp 268–290Google Scholar
  31. 31.
    Lyapunov AA (1940) O vpolnye additivnykh vyektor-funktsiyakh: // Izvyestiya Akadyemii nauk SSSR. Syer. matyematichyeskaya. 4(6):465–478Google Scholar
  32. 32.
    Lyusternik LA (1934) On constrained exstrema of functionals (in Rusian). Matem. Sbornik 41(3):390–401. See also Russian Math Surv 35(6):11–51 (1980)Google Scholar
  33. 33.
    Mayer A (1886) Begründung der Lagrangesche Multiplikatorenmethode der Variatinsrehbung. Math Ann 26Google Scholar
  34. 34.
    Minkovski H (1910) Geometrie der Zahlen. Teubner, LeipzigGoogle Scholar
  35. 35.
    Monge G (1781) Memoire sur la théorie des déblais et des remblais, ParisGoogle Scholar
  36. 36.
    Moreau JJ (1964) Fonctionelles sus-differènciables. C R Acad Sci (Paris) 258:1128–1931MathSciNetGoogle Scholar
  37. 37.
    Newman DJ (1965) Location of the maximum on unimodal surfices. J ACM 12 No 3:395–398Google Scholar
  38. 38.
    Newton I (1999) Mathematical principles of natural philosophy. University of California Press, BerkeleyMATHGoogle Scholar
  39. 39.
    Newton I, Whiteside DT (1967–1982). The mathematical papers of Isaac Newton, 8 vols. Cambridge University Press, Cambridge. ISBN:0-521-07740-0Google Scholar
  40. 40.
    Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes (Russian). English translation: InterscienceGoogle Scholar
  41. 41.
    Rockafellar RT (1997) Convex analysis. Princeton landmarks in mathematics. Princeton University Press, PrincetonMATHGoogle Scholar
  42. 42.
    Weierstrass K (1927) Mathematische Werke, Bd 7. Vorlesungemn uber Variationsrehtung. Akad. Verlag, Berlin–LeipzigGoogle Scholar
  43. 43.
    Avakov ER, Magaril-Il’yaev GG, Tikhomirov VM (2013) Lagrange’s principle in extremum problems with constraints. Usp Mat Nauk 68(3):5–38MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Mech-math department of Lomonosov Moscow State UniversityMoscowRussia

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