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.
JEL Classification:
Mathematics Subject Classification (2010): 26B10, 26B25, 34A55, 49-02, 49J15, 49K15, 90C25
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Euclid (2007) Euclid’s elements. Green Lion Press, Ann Arbor
Alexeev V, Tikhomirov V, Fomin S (1987) Optimal control. Plenum Publishing Corporation, New York
Banach S (1932) “Théorie des opérations linéaire”, Warszava, Monographje Matematyczne
Bernoulli I (1696) Problema novum, ad cujus solutionem Matematici invitantur. Acta Eruditorum
Bliss GA (1963) Lectures on the calculus of variations. University of Chicago Press
de Fermat P (1891) Oeuvres de Fermat, vol 1. Gauthier-Villars, Paris
Dini U (1877/1878) Analisi infinitesimale. Lezzione dettate nella Università Pisa. Bd 2
Dmitruk AV, Milyutin AA, Osmolovskii NP (1980) Lyustrenik’s theorem and the theory of extrema. Uspehi Mat Nauk 35(6): 11–46
Dantzig GB (1963) Linear Programming and Extensions. Princeton University Press, Princeton, NJ
Dubovitskii AY, Milyutin AA (1965) Extremum problems with constraints. Zh Vychisl Mat i Mat Fiz 5:395–453
Euler L (1744) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latssimo sencu accepti. Lausanne
Fenchel W (1953) Convex Cones, Sets, and Functions. Princeton University Department of Mathematics, Princeton, NJ
Frèchet V (1912) Sur la notion de differentielle. Nouvelle annale de mathematique, Ser. 4, V.XII. S. 845
Graves LM (1950) Some mapping theorems. Duke Math J 17:111–114
Hamilton WR (1835) Second essay on a general methods in dynamics. Philos Trans R Soc Pt 1:95–144
Hilbert D. Hazewinkel M (ed) (2001) “Hilbert problems”, Encyclopedia of mathematics. Springer. ISBN:978-1-55608-010-4
Ioffe A, Tikhomirov V (1979) Theory of extremal problems. North-Holland, Amsterdam
Jacobi CGJ (1837) Zur Theorie der Variations-Rechnung und der Differential-Gleichungen. Krelle’s Journall 17:68–82
John F Extremal problems with inequalities as subsidery conditions. In: Studies and Essays. Courant Anniverrsary Volume, 1948, pp 187–204
Kantorovich LV (1939) Mathematical methods of organizing and planning production. Manag Sci 6(4)(Jul., 1960):366–422
Karush WE (1939) Minima of functions of several variables with inequalities as side conditions, University of Chicago Press
Kepler I (1615) The volume of a Wine Barrel – Kepler’s Nova stereometria doliorum vinariorum, Lincii (Roberto Cardil Matematicas Visuales)
Kneser A (1925) Lehrbuch der Variationsrechnung. Springer, Wiesbaden
Kuhn HW, Tucker AW (1951) Nonlinear programming. University of California Press, Berkley, pp 481–482
Lagrange JL (1766) Essai d’une nouvelle méthode pour determiner les maxima et les minima periales Petropolitanae, vol 10, 51–93
Lagrange JL (1797) Théorie des fonctions analytiques, Paris
Leach E (1961) A note on inverse mapping theorem. Proc AMS 12:694–697
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. 7–37
Leibniz G (1684) Acta Eroditorum, L.M.S., t. V, pp 220–226
Levin AY On an algorithm for the minimization of convex function Sov. Math., Dokl. – 1965. – no. 6, pp 268–290
Lyapunov AA (1940) O vpolnye additivnykh vyektor-funktsiyakh: // Izvyestiya Akadyemii nauk SSSR. Syer. matyematichyeskaya. 4(6):465–478
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)
Mayer A (1886) Begründung der Lagrangesche Multiplikatorenmethode der Variatinsrehbung. Math Ann 26
Minkovski H (1910) Geometrie der Zahlen. Teubner, Leipzig
Monge G (1781) Memoire sur la théorie des déblais et des remblais, Paris
Moreau JJ (1964) Fonctionelles sus-differènciables. C R Acad Sci (Paris) 258:1128–1931
Newman DJ (1965) Location of the maximum on unimodal surfices. J ACM 12 No 3:395–398
Newton I (1999) Mathematical principles of natural philosophy. University of California Press, Berkeley
Newton I, Whiteside DT (1967–1982). The mathematical papers of Isaac Newton, 8 vols. Cambridge University Press, Cambridge. ISBN:0-521-07740-0
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes (Russian). English translation: Interscience
Rockafellar RT (1997) Convex analysis. Princeton landmarks in mathematics. Princeton University Press, Princeton
Weierstrass K (1927) Mathematische Werke, Bd 7. Vorlesungemn uber Variationsrehtung. Akad. Verlag, Berlin–Leipzig
Avakov ER, Magaril-Il’yaev GG, Tikhomirov VM (2013) Lagrange’s principle in extremum problems with constraints. Usp Mat Nauk 68(3):5–38
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media Singapore
About this chapter
Cite this chapter
Tikhomirov, V. (2016). Survey of the Theory of Extremal Problems. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 20. Advances in Mathematical Economics, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-10-0476-6_6
Download citation
DOI: https://doi.org/10.1007/978-981-10-0476-6_6
Received:
Revised:
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-0475-9
Online ISBN: 978-981-10-0476-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)