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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  • $begingroup$
    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.
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    – M. Winter
    Jan 10 at 21:24






  • 1




    $begingroup$
    mathoverflow.net/q/252206/12674 looks relevant.
    $endgroup$
    – Thomas Kalinowski
    Jan 11 at 2:04















10












$begingroup$


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











$endgroup$











  • $begingroup$
    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.
    $endgroup$
    – M. Winter
    Jan 10 at 21:24






  • 1




    $begingroup$
    mathoverflow.net/q/252206/12674 looks relevant.
    $endgroup$
    – Thomas Kalinowski
    Jan 11 at 2:04













10












10








10


5



$begingroup$


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











$endgroup$




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 Jan 10 at 20:06


























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Tom Solberg












  • $begingroup$
    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.
    $endgroup$
    – M. Winter
    Jan 10 at 21:24






  • 1




    $begingroup$
    mathoverflow.net/q/252206/12674 looks relevant.
    $endgroup$
    – Thomas Kalinowski
    Jan 11 at 2:04
















  • $begingroup$
    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.
    $endgroup$
    – M. Winter
    Jan 10 at 21:24






  • 1




    $begingroup$
    mathoverflow.net/q/252206/12674 looks relevant.
    $endgroup$
    – Thomas Kalinowski
    Jan 11 at 2:04















$begingroup$
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.
$endgroup$
– M. Winter
Jan 10 at 21:24




$begingroup$
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.
$endgroup$
– M. Winter
Jan 10 at 21:24




1




1




$begingroup$
mathoverflow.net/q/252206/12674 looks relevant.
$endgroup$
– Thomas Kalinowski
Jan 11 at 2:04




$begingroup$
mathoverflow.net/q/252206/12674 looks relevant.
$endgroup$
– Thomas Kalinowski
Jan 11 at 2:04










3 Answers
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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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    3












    $begingroup$

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






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    $endgroup$




















      2












      $begingroup$

      Bondareva–Shapley_theorem is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron.






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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        5












        $begingroup$

        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











        $endgroup$

















          5












          $begingroup$

          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











          $endgroup$















            5












            5








            5





            $begingroup$

            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











            $endgroup$



            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 Jan 10 at 23:03


























            community wiki





            Timothy Chow






















                3












                $begingroup$

                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











                $endgroup$

















                  3












                  $begingroup$

                  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











                  $endgroup$















                    3












                    3








                    3





                    $begingroup$

                    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











                    $endgroup$



                    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 Jan 11 at 2:46


























                    community wiki





                    Thomas Kalinowski






















                        2












                        $begingroup$

                        Bondareva–Shapley_theorem is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron.






                        share|cite|improve this answer











                        $endgroup$

















                          2












                          $begingroup$

                          Bondareva–Shapley_theorem is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron.






                          share|cite|improve this answer











                          $endgroup$















                            2












                            2








                            2





                            $begingroup$

                            Bondareva–Shapley_theorem is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron.






                            share|cite|improve this answer











                            $endgroup$



                            Bondareva–Shapley_theorem is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron.







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            answered Jan 11 at 5:43


























                            community wiki





                            Fedor Petrov




























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