Saddle point and upper and lower value
Clash Royale CLAN TAG#URR8PPP
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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
asked Aug 23 at 8:06
SHAN
1208
add a comment |Â
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
asked Aug 23 at 8:06
SHAN
1208
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
add a comment |Â
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
asked Aug 23 at 8:06
SHAN
1208
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
asked Aug 23 at 8:06
SHAN
1208
asked Aug 23 at 8:06
SHAN
1208
asked Aug 23 at 8:06
SHAN
1208
asked Aug 23 at 8:06
SHAN
1208
1208
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
add a comment |Â
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
add a comment |Â
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 $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.
answered Aug 23 at 8:29
mathcounterexamples.net
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
add a comment |Â
1 Answer
1
active
oldest
votes
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 $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.
answered Aug 23 at 8:29
mathcounterexamples.net
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
add a comment |Â
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 $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.
answered Aug 23 at 8:29
mathcounterexamples.net
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
add a comment |Â
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 $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.
answered Aug 23 at 8:29
mathcounterexamples.net
25.5k21755
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 $inf_xsup_yf(x,y)=sup_yinf_xf(x,y)$.
answered Aug 23 at 8:29
mathcounterexamples.net
25.5k21755
edited Aug 23 at 8:55
answered Aug 23 at 8:29
mathcounterexamples.net
25.5k21755
answered Aug 23 at 8:29
mathcounterexamples.net
25.5k21755
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
add a comment |Â
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
add a comment |Â
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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