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Farkaseva lema: elementarni dokaz i ekonomske primjene

Author

Listed:
  • Margareta Gardijan

    (Faculty of Economics and Business, University of Zagreb)

  • Vedran Kojić

    (Faculty of Economics and Business, University of Zagreb)

  • Marina Slišković

    (Faculty of Economics and Business, University of Zagreb)

Abstract

U ovom radu donosimo elementarni dokaz Farkaseve leme. U matematici je Farkaseva lema vrlo bitna činjenica koja se koristi u teoriji optimizacije, primjerice u izvođenju Karush-Khun-Tuckerovih uvjeta optimalnosti u slučaju ograničenja u obliku nejednakosti kod nelinearnog programiranja, te u dokazivanju dualnih teorema za linearno programiranje. Iako je Farkasevu lemu vrlo jednostavno iskazati, njezin dokaz nije trivijalan (većina dokaza se temelji na netrivijalnim rezultatima iz područja optimizacije i (linearne) algebre), o čemu govori i podatak da su ju mnogi na različite načine dokazivali još od 1972. (pa i ranije) sve do danas, nadmećući se pritom tko će ponuditi jednostavniji dokaz. U ovom radu Farkasevu lemu dokazujemo na elementaran način koristeći matematičku indukciju. Dokaz ove leme matematičkom indukcijom je poznat u stranoj, ali ne i u domaćoj literaturi. Stoga je cilj ovog rada revidirati taj dokaz, ispraviti postojeće nedostatke i pogreške, te detaljno objasniti svaku stavku dokaza, nekoristeći pritom složene termine i činjenice iz područja optimizacije i algebre. Osim samog dokaza Farkaseve leme, navodimo i njezine dvije primjene u ekonomiji, čime želimo, s jedne strane, približiti i objasniti Farkasevu lemu na razumljiv način čitateljima koji po svom zvanju nisu matematičari, ali ju koriste u svom radu, te s druge strane doprinijeti razumijevanju samog iskaza Farkaseve leme kroz konkretne primjere.

Suggested Citation

  • Margareta Gardijan & Vedran Kojić & Marina Slišković, 2018. "Farkaseva lema: elementarni dokaz i ekonomske primjene," EFZG Working Papers Series 1801, Faculty of Economics and Business, University of Zagreb.
    • Handle: RePEc:zag:wpaper:1801
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    More about this item

    Keywords

    Farkaseva lema; matematički dokaz; matematička indukcija; financijsko modeliranje; teorija igara;
    All these keywords.

    JEL classification:

    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)

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