Saddle Point Maximization
$begingroup$
My question is that in general, is there a case where saddle point be the global max of a function?
I am solving a game theory question which the optimal solution is the saddle point. Can I conclude that the optimal solution is at the boundaries?
optimization
asked Jan 4 at 18:30
user3425989user3425989
11
$endgroup$
add a comment |
$begingroup$
My question is that in general, is there a case where saddle point be the global max of a function?
I am solving a game theory question which the optimal solution is the saddle point. Can I conclude that the optimal solution is at the boundaries?
optimization
asked Jan 4 at 18:30
user3425989user3425989
11
$endgroup$
$begingroup$
I thought saddles were by definition neither max nor min.
$endgroup$
– Randall
Jan 4 at 18:32
$begingroup$
What game theory question?
$endgroup$
– cgiovanardi
Jan 5 at 0:02
add a comment |
$begingroup$
My question is that in general, is there a case where saddle point be the global max of a function?
I am solving a game theory question which the optimal solution is the saddle point. Can I conclude that the optimal solution is at the boundaries?
optimization
asked Jan 4 at 18:30
user3425989user3425989
11
$endgroup$
My question is that in general, is there a case where saddle point be the global max of a function?
I am solving a game theory question which the optimal solution is the saddle point. Can I conclude that the optimal solution is at the boundaries?
optimization
optimization
asked Jan 4 at 18:30
user3425989user3425989
11
asked Jan 4 at 18:30
user3425989user3425989
11
asked Jan 4 at 18:30
user3425989user3425989
11
asked Jan 4 at 18:30
user3425989user3425989
11
asked Jan 4 at 18:30
user3425989user3425989
11
11
$begingroup$
I thought saddles were by definition neither max nor min.
$endgroup$
– Randall
Jan 4 at 18:32
$begingroup$
What game theory question?
$endgroup$
– cgiovanardi
Jan 5 at 0:02
add a comment |
$begingroup$
I thought saddles were by definition neither max nor min.
$endgroup$
– Randall
Jan 4 at 18:32
$begingroup$
What game theory question?
$endgroup$
– cgiovanardi
Jan 5 at 0:02
$begingroup$
I thought saddles were by definition neither max nor min.
$endgroup$
– Randall
Jan 4 at 18:32
$begingroup$
I thought saddles were by definition neither max nor min.
$endgroup$
– Randall
Jan 4 at 18:32
$begingroup$
What game theory question?
$endgroup$
– cgiovanardi
Jan 5 at 0:02
$begingroup$
What game theory question?
$endgroup$
– cgiovanardi
Jan 5 at 0:02
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
By definition, a saddle point is
- a local min in some directions and
- a local max in other directions at the same time.
Since it is a local min in at least one direction, there are more optimal points for maximization. Ditto minimization from the other directions...
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
$endgroup$
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
add a comment |
$begingroup$
It is a common misunderstanding in optimization how a saddle point may be the optimal point. The saddle point that is the optimal solution is the saddle point for the Lagrange function $L(x,u,v)$, not the saddle point for the objective function itself.
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
By definition, a saddle point is
- a local min in some directions and
- a local max in other directions at the same time.
Since it is a local min in at least one direction, there are more optimal points for maximization. Ditto minimization from the other directions...
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
$endgroup$
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
add a comment |
$begingroup$
By definition, a saddle point is
- a local min in some directions and
- a local max in other directions at the same time.
Since it is a local min in at least one direction, there are more optimal points for maximization. Ditto minimization from the other directions...
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
$endgroup$
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
add a comment |
$begingroup$
By definition, a saddle point is
- a local min in some directions and
- a local max in other directions at the same time.
Since it is a local min in at least one direction, there are more optimal points for maximization. Ditto minimization from the other directions...
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
$endgroup$
By definition, a saddle point is
- a local min in some directions and
- a local max in other directions at the same time.
Since it is a local min in at least one direction, there are more optimal points for maximization. Ditto minimization from the other directions...
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
answered Jan 4 at 18:32
gt6989bgt6989b
34.6k22456
34.6k22456
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
add a comment |
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
Thanks, So can we say that is is only boundary solutions?
$endgroup$
– user3425989
Jan 4 at 18:40
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
$begingroup$
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary.
$endgroup$
– gt6989b
Jan 6 at 5:29
add a comment |
$begingroup$
It is a common misunderstanding in optimization how a saddle point may be the optimal point. The saddle point that is the optimal solution is the saddle point for the Lagrange function $L(x,u,v)$, not the saddle point for the objective function itself.
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
$endgroup$
add a comment |
$begingroup$
It is a common misunderstanding in optimization how a saddle point may be the optimal point. The saddle point that is the optimal solution is the saddle point for the Lagrange function $L(x,u,v)$, not the saddle point for the objective function itself.
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
$endgroup$
add a comment |
$begingroup$
It is a common misunderstanding in optimization how a saddle point may be the optimal point. The saddle point that is the optimal solution is the saddle point for the Lagrange function $L(x,u,v)$, not the saddle point for the objective function itself.
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
$endgroup$
It is a common misunderstanding in optimization how a saddle point may be the optimal point. The saddle point that is the optimal solution is the saddle point for the Lagrange function $L(x,u,v)$, not the saddle point for the objective function itself.
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
answered Jan 4 at 18:39
A.Γ.A.Γ.
22.8k32656
22.8k32656
add a comment |
add a comment |
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$begingroup$
I thought saddles were by definition neither max nor min.
$endgroup$
– Randall
Jan 4 at 18:32
$begingroup$
What game theory question?
$endgroup$
– cgiovanardi
Jan 5 at 0:02