Saddle point and upper and lower value

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Suppose $f(x,y)$ is a function defined on $mathbb Rtimesmathbb R$, it is easy to see that $$inf_xsup_yf(x,y)geqsup_yinf_xf(x,y).$$ Moreover, if
there exists a pair $(x^*,y^*)$ such that for all $x,yinmathbb R$,$$f(x^*,y)leq f(x^*,y^*)leq f(x,y^*),$$
then $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y).$$
Is the existence of a saddle point $(x^*,y^*)$ also a necessary condition of $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y)?$$ Thanks!




real-analysis




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  • For the sufficiency part, there is no restriction on the range of $x$ and $y$. For the necessity part, I am not sure the condition on the range of $x$ and $y$.
    – SHAN
    Aug 23 at 9:06














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1
down vote

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Suppose $f(x,y)$ is a function defined on $mathbb Rtimesmathbb R$, it is easy to see that $$inf_xsup_yf(x,y)geqsup_yinf_xf(x,y).$$ Moreover, if
there exists a pair $(x^*,y^*)$ such that for all $x,yinmathbb R$,$$f(x^*,y)leq f(x^*,y^*)leq f(x,y^*),$$
then $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y).$$
Is the existence of a saddle point $(x^*,y^*)$ also a necessary condition of $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y)?$$ Thanks!




real-analysis




share|cite|improve this question




















  • For the sufficiency part, there is no restriction on the range of $x$ and $y$. For the necessity part, I am not sure the condition on the range of $x$ and $y$.
    – SHAN
    Aug 23 at 9:06












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Suppose $f(x,y)$ is a function defined on $mathbb Rtimesmathbb R$, it is easy to see that $$inf_xsup_yf(x,y)geqsup_yinf_xf(x,y).$$ Moreover, if
there exists a pair $(x^*,y^*)$ such that for all $x,yinmathbb R$,$$f(x^*,y)leq f(x^*,y^*)leq f(x,y^*),$$
then $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y).$$
Is the existence of a saddle point $(x^*,y^*)$ also a necessary condition of $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y)?$$ Thanks!




real-analysis




share|cite|improve this question












Suppose $f(x,y)$ is a function defined on $mathbb Rtimesmathbb R$, it is easy to see that $$inf_xsup_yf(x,y)geqsup_yinf_xf(x,y).$$ Moreover, if
there exists a pair $(x^*,y^*)$ such that for all $x,yinmathbb R$,$$f(x^*,y)leq f(x^*,y^*)leq f(x,y^*),$$
then $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y).$$
Is the existence of a saddle point $(x^*,y^*)$ also a necessary condition of $$inf_xsup_yf(x,y)=sup_yinf_xf(x,y)?$$ Thanks!





real-analysis





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asked Aug 23 at 8:06









SHAN

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  • For the sufficiency part, there is no restriction on the range of $x$ and $y$. For the necessity part, I am not sure the condition on the range of $x$ and $y$.
    – SHAN
    Aug 23 at 9:06
















  • For the sufficiency part, there is no restriction on the range of $x$ and $y$. For the necessity part, I am not sure the condition on the range of $x$ and $y$.
    – SHAN
    Aug 23 at 9:06















For the sufficiency part, there is no restriction on the range of $x$ and $y$. For the necessity part, I am not sure the condition on the range of $x$ and $y$.
– SHAN
Aug 23 at 9:06




For the sufficiency part, there is no restriction on the range of $x$ and $y$. For the necessity part, I am not sure the condition on the range of $x$ and $y$.
– SHAN
Aug 23 at 9:06










1 Answer
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Hint: the answer is positive



The answer is indeed positive. Unfortunately, the only reference I have is in French: Wikipedia - Point col. The English version of the article provides with less content.



The precise result mentioned in that article is:



Characterization of saddle points: a couple $(x^*,y^*) in X times Y$ is a saddle point for $f$ if and only if $x^*$ is solution of $inf_xsup_yf(x,y)$ (primal problem), $y^*$ is solution of $sup_yinf_xf(x,y)$ and $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.






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  • In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
    – SHAN
    Aug 23 at 9:01











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













Hint: the answer is positive



The answer is indeed positive. Unfortunately, the only reference I have is in French: Wikipedia - Point col. The English version of the article provides with less content.



The precise result mentioned in that article is:



Characterization of saddle points: a couple $(x^*,y^*) in X times Y$ is a saddle point for $f$ if and only if $x^*$ is solution of $inf_xsup_yf(x,y)$ (primal problem), $y^*$ is solution of $sup_yinf_xf(x,y)$ and $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.






share|cite|improve this answer






















  • In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
    – SHAN
    Aug 23 at 9:01















up vote
0
down vote













Hint: the answer is positive



The answer is indeed positive. Unfortunately, the only reference I have is in French: Wikipedia - Point col. The English version of the article provides with less content.



The precise result mentioned in that article is:



Characterization of saddle points: a couple $(x^*,y^*) in X times Y$ is a saddle point for $f$ if and only if $x^*$ is solution of $inf_xsup_yf(x,y)$ (primal problem), $y^*$ is solution of $sup_yinf_xf(x,y)$ and $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.






share|cite|improve this answer






















  • In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
    – SHAN
    Aug 23 at 9:01













up vote
0
down vote










up vote
0
down vote









Hint: the answer is positive



The answer is indeed positive. Unfortunately, the only reference I have is in French: Wikipedia - Point col. The English version of the article provides with less content.



The precise result mentioned in that article is:



Characterization of saddle points: a couple $(x^*,y^*) in X times Y$ is a saddle point for $f$ if and only if $x^*$ is solution of $inf_xsup_yf(x,y)$ (primal problem), $y^*$ is solution of $sup_yinf_xf(x,y)$ and $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.






share|cite|improve this answer














Hint: the answer is positive



The answer is indeed positive. Unfortunately, the only reference I have is in French: Wikipedia - Point col. The English version of the article provides with less content.



The precise result mentioned in that article is:



Characterization of saddle points: a couple $(x^*,y^*) in X times Y$ is a saddle point for $f$ if and only if $x^*$ is solution of $inf_xsup_yf(x,y)$ (primal problem), $y^*$ is solution of $sup_yinf_xf(x,y)$ and $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.







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edited Aug 23 at 8:55

























answered Aug 23 at 8:29









mathcounterexamples.net

25.5k21755




25.5k21755











  • In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
    – SHAN
    Aug 23 at 9:01

















  • In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
    – SHAN
    Aug 23 at 9:01
















In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
– SHAN
Aug 23 at 9:01





In the English version, the content of Characterization of stool-points is: A couple of points $(bar x, bar y)$ is a saddle point of $f$ on $X times Y$ if and only if, $bar x$ is solution of the primal problem ( P ) , $bar y$ is the solution of the dual problem ( D ) and there is no duality jump. So does this mean the existence of saddle point is only a sufficient condition of no duality jump?
– SHAN
Aug 23 at 9:01


















 

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