Famous theorems that are special cases of linear programming (or convex) duality












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The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.










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  • The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
    – M. Winter
    8 hours ago






  • 1




    mathoverflow.net/q/252206/12674 looks relevant.
    – Thomas Kalinowski
    3 hours ago
















5














The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.










share|cite|improve this question
























  • The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
    – M. Winter
    8 hours ago






  • 1




    mathoverflow.net/q/252206/12674 looks relevant.
    – Thomas Kalinowski
    3 hours ago














5












5








5


1





The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.










share|cite|improve this question















The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.







oc.optimization-and-control convex-optimization linear-programming






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asked 9 hours ago


























community wiki





Tom Solberg













  • The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
    – M. Winter
    8 hours ago






  • 1




    mathoverflow.net/q/252206/12674 looks relevant.
    – Thomas Kalinowski
    3 hours ago


















  • The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
    – M. Winter
    8 hours ago






  • 1




    mathoverflow.net/q/252206/12674 looks relevant.
    – Thomas Kalinowski
    3 hours ago
















The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago




The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago




1




1




mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago




mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago










2 Answers
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4














To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.






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    2














    Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).






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      2 Answers
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      2 Answers
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      active

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      4














      To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.






      share|cite|improve this answer




























        4














        To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.






        share|cite|improve this answer


























          4












          4








          4






          To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.






          share|cite|improve this answer














          To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          answered 6 hours ago


























          community wiki





          Timothy Chow
























              2














              Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).






              share|cite|improve this answer




























                2














                Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).






                share|cite|improve this answer


























                  2












                  2








                  2






                  Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).






                  share|cite|improve this answer














                  Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  answered 2 hours ago


























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                  Thomas Kalinowski































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