Mixing
Minimization of function with two parameters.
Clash Royale CLAN TAG#URR8PPP
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Let this minimization,
$$
min_xin[-2y,4y]((1-(y+frac12x))^++(5/4-(y+frac14x))^++(7/4-(y-frac14x))^+) (*)
$$
with $ygeq 0$, $xinmathbb R$ and the notation $(.)^+$ means :
$(z)^+=max(0,z)$.
Is there a solution of $(*)$ as in this example :
$$
min_xin[-y/2,y] [(-(y+2x))^++(-(y+x))^++(1-(y-x))^+]=(1-3/2y)^+
$$
Where the minimum is attained at $x = -y/2$.
I have tried by the mean of differentiation without success.
Is $x=-2y$ the solution of $(*)$ ? or it depends to $y$ ?
real-analysis functional-analysis optimization convex-optimization
asked Jul 25 at 15:25
the-owner
11410
add a comment |Â
up vote
0
down vote
favorite
Let this minimization,
$$
min_xin[-2y,4y]((1-(y+frac12x))^++(5/4-(y+frac14x))^++(7/4-(y-frac14x))^+) (*)
$$
with $ygeq 0$, $xinmathbb R$ and the notation $(.)^+$ means :
$(z)^+=max(0,z)$.
Is there a solution of $(*)$ as in this example :
$$
min_xin[-y/2,y] [(-(y+2x))^++(-(y+x))^++(1-(y-x))^+]=(1-3/2y)^+
$$
Where the minimum is attained at $x = -y/2$.
I have tried by the mean of differentiation without success.
Is $x=-2y$ the solution of $(*)$ ? or it depends to $y$ ?
real-analysis functional-analysis optimization convex-optimization
asked Jul 25 at 15:25
the-owner
11410
why don't you reformulate it as a linear optimization problem and see if you can find a solution to the KKT conditions?
– LinAlg
Jul 25 at 17:45
@LinAlg I am not familiar with the KKT conditions and linear optimization ...
– the-owner
Jul 25 at 18:51
then just create a few plots for different values of $y$ and see if you can detect a pattern :)
– LinAlg
Jul 25 at 19:16
What the notation $(cdot ) ^+$ means?
– Cesareo
Jul 25 at 23:32
@Cesareo $(z)^+=z$ if $z>0$ and $(z)^+=0$ if $zleq 0$ in other words : $(z)^+=zmathbb I_z>0$
– the-owner
Jul 26 at 5:36
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let this minimization,
$$
min_xin[-2y,4y]((1-(y+frac12x))^++(5/4-(y+frac14x))^++(7/4-(y-frac14x))^+) (*)
$$
with $ygeq 0$, $xinmathbb R$ and the notation $(.)^+$ means :
$(z)^+=max(0,z)$.
Is there a solution of $(*)$ as in this example :
$$
min_xin[-y/2,y] [(-(y+2x))^++(-(y+x))^++(1-(y-x))^+]=(1-3/2y)^+
$$
Where the minimum is attained at $x = -y/2$.
I have tried by the mean of differentiation without success.
Is $x=-2y$ the solution of $(*)$ ? or it depends to $y$ ?
real-analysis functional-analysis optimization convex-optimization
asked Jul 25 at 15:25
the-owner
11410
Let this minimization,
$$
min_xin[-2y,4y]((1-(y+frac12x))^++(5/4-(y+frac14x))^++(7/4-(y-frac14x))^+) (*)
$$
with $ygeq 0$, $xinmathbb R$ and the notation $(.)^+$ means :
$(z)^+=max(0,z)$.
Is there a solution of $(*)$ as in this example :
$$
min_xin[-y/2,y] [(-(y+2x))^++(-(y+x))^++(1-(y-x))^+]=(1-3/2y)^+
$$
Where the minimum is attained at $x = -y/2$.
I have tried by the mean of differentiation without success.
Is $x=-2y$ the solution of $(*)$ ? or it depends to $y$ ?
real-analysis functional-analysis optimization convex-optimization
asked Jul 25 at 15:25
the-owner
11410
edited Jul 26 at 9:23
asked Jul 25 at 15:25
the-owner
11410
asked Jul 25 at 15:25
the-owner
11410
asked Jul 25 at 15:25
the-owner
11410
11410
why don't you reformulate it as a linear optimization problem and see if you can find a solution to the KKT conditions?
– LinAlg
Jul 25 at 17:45
@LinAlg I am not familiar with the KKT conditions and linear optimization ...
– the-owner
Jul 25 at 18:51
then just create a few plots for different values of $y$ and see if you can detect a pattern :)
– LinAlg
Jul 25 at 19:16
What the notation $(cdot ) ^+$ means?
– Cesareo
Jul 25 at 23:32
@Cesareo $(z)^+=z$ if $z>0$ and $(z)^+=0$ if $zleq 0$ in other words : $(z)^+=zmathbb I_z>0$
– the-owner
Jul 26 at 5:36
add a comment |Â
why don't you reformulate it as a linear optimization problem and see if you can find a solution to the KKT conditions?
– LinAlg
Jul 25 at 17:45
@LinAlg I am not familiar with the KKT conditions and linear optimization ...
– the-owner
Jul 25 at 18:51
then just create a few plots for different values of $y$ and see if you can detect a pattern :)
– LinAlg
Jul 25 at 19:16
What the notation $(cdot ) ^+$ means?
– Cesareo
Jul 25 at 23:32
@Cesareo $(z)^+=z$ if $z>0$ and $(z)^+=0$ if $zleq 0$ in other words : $(z)^+=zmathbb I_z>0$
– the-owner
Jul 26 at 5:36
why don't you reformulate it as a linear optimization problem and see if you can find a solution to the KKT conditions?
– LinAlg
Jul 25 at 17:45
why don't you reformulate it as a linear optimization problem and see if you can find a solution to the KKT conditions?
– LinAlg
Jul 25 at 17:45
@LinAlg I am not familiar with the KKT conditions and linear optimization ...
– the-owner
Jul 25 at 18:51
@LinAlg I am not familiar with the KKT conditions and linear optimization ...
– the-owner
Jul 25 at 18:51
then just create a few plots for different values of $y$ and see if you can detect a pattern :)
– LinAlg
Jul 25 at 19:16
then just create a few plots for different values of $y$ and see if you can detect a pattern :)
– LinAlg
Jul 25 at 19:16
What the notation $(cdot ) ^+$ means?
– Cesareo
Jul 25 at 23:32
What the notation $(cdot ) ^+$ means?
– Cesareo
Jul 25 at 23:32
@Cesareo $(z)^+=z$ if $z>0$ and $(z)^+=0$ if $zleq 0$ in other words : $(z)^+=zmathbb I_z>0$
– the-owner
Jul 26 at 5:36
@Cesareo $(z)^+=z$ if $z>0$ and $(z)^+=0$ if $zleq 0$ in other words : $(z)^+=zmathbb I_z>0$
– the-owner
Jul 26 at 5:36
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
This problem can be arranged as a LP program
$$
min_x,y,z_1,z_2,z_3 z_1+z_2+z_3 ;;mboxs. t.
$$
$$
z_1=1-y-frac x2\
z_2=frac 54-y-frac x4\
z_3=frac 74-y+frac x4\
z_1 ge 0\
z_2 ge 0\
z_3 ge 0\
-2y le x\
xle 4y\
y ge 0
$$
with one solution at
$$
x = -1, y = frac 32, z_1 = 0, z_2 = 0, z_3 = 0
$$
Attached a plot showing in black the level surfaces for $f(x,y) = max( 0,z_1)+max(0,z_2)+max(0,z_3);$ in red the lines $z_1=0, z_2=0, z_3=0$
answered Jul 26 at 8:54
Cesareo
5,6472412
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
This problem can be arranged as a LP program
$$
min_x,y,z_1,z_2,z_3 z_1+z_2+z_3 ;;mboxs. t.
$$
$$
z_1=1-y-frac x2\
z_2=frac 54-y-frac x4\
z_3=frac 74-y+frac x4\
z_1 ge 0\
z_2 ge 0\
z_3 ge 0\
-2y le x\
xle 4y\
y ge 0
$$
with one solution at
$$
x = -1, y = frac 32, z_1 = 0, z_2 = 0, z_3 = 0
$$
Attached a plot showing in black the level surfaces for $f(x,y) = max( 0,z_1)+max(0,z_2)+max(0,z_3);$ in red the lines $z_1=0, z_2=0, z_3=0$
answered Jul 26 at 8:54
Cesareo
5,6472412
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
add a comment |Â
up vote
2
down vote
This problem can be arranged as a LP program
$$
min_x,y,z_1,z_2,z_3 z_1+z_2+z_3 ;;mboxs. t.
$$
$$
z_1=1-y-frac x2\
z_2=frac 54-y-frac x4\
z_3=frac 74-y+frac x4\
z_1 ge 0\
z_2 ge 0\
z_3 ge 0\
-2y le x\
xle 4y\
y ge 0
$$
with one solution at
$$
x = -1, y = frac 32, z_1 = 0, z_2 = 0, z_3 = 0
$$
Attached a plot showing in black the level surfaces for $f(x,y) = max( 0,z_1)+max(0,z_2)+max(0,z_3);$ in red the lines $z_1=0, z_2=0, z_3=0$
answered Jul 26 at 8:54
Cesareo
5,6472412
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
add a comment |Â
up vote
2
down vote
up vote
2
down vote
This problem can be arranged as a LP program
$$
min_x,y,z_1,z_2,z_3 z_1+z_2+z_3 ;;mboxs. t.
$$
$$
z_1=1-y-frac x2\
z_2=frac 54-y-frac x4\
z_3=frac 74-y+frac x4\
z_1 ge 0\
z_2 ge 0\
z_3 ge 0\
-2y le x\
xle 4y\
y ge 0
$$
with one solution at
$$
x = -1, y = frac 32, z_1 = 0, z_2 = 0, z_3 = 0
$$
Attached a plot showing in black the level surfaces for $f(x,y) = max( 0,z_1)+max(0,z_2)+max(0,z_3);$ in red the lines $z_1=0, z_2=0, z_3=0$
answered Jul 26 at 8:54
Cesareo
5,6472412
This problem can be arranged as a LP program
$$
min_x,y,z_1,z_2,z_3 z_1+z_2+z_3 ;;mboxs. t.
$$
$$
z_1=1-y-frac x2\
z_2=frac 54-y-frac x4\
z_3=frac 74-y+frac x4\
z_1 ge 0\
z_2 ge 0\
z_3 ge 0\
-2y le x\
xle 4y\
y ge 0
$$
with one solution at
$$
x = -1, y = frac 32, z_1 = 0, z_2 = 0, z_3 = 0
$$
Attached a plot showing in black the level surfaces for $f(x,y) = max( 0,z_1)+max(0,z_2)+max(0,z_3);$ in red the lines $z_1=0, z_2=0, z_3=0$
answered Jul 26 at 8:54
Cesareo
5,6472412
answered Jul 26 at 8:54
Cesareo
5,6472412
answered Jul 26 at 8:54
Cesareo
5,6472412
answered Jul 26 at 8:54
Cesareo
5,6472412
5,6472412
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
add a comment |Â
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
Your post is interesting but could we obtain a solution (for the min) on $xin[-2y,4y]$ for any $y>0$ not a single point (x,y) but something more general ? for instance $y<alpha$ then $x=beta y$ and $ygeq alpha$ then $x=gamma y$ (it is just an example like in my post I have provided another one). Do you understand my idea ? It is maybe the bold blue line what I am looking for ? it is something like $max(0,a+b.y)$ ?
– the-owner
Jul 26 at 9:10
add a comment |Â
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why don't you reformulate it as a linear optimization problem and see if you can find a solution to the KKT conditions?
– LinAlg
Jul 25 at 17:45
@LinAlg I am not familiar with the KKT conditions and linear optimization ...
– the-owner
Jul 25 at 18:51
then just create a few plots for different values of $y$ and see if you can detect a pattern :)
– LinAlg
Jul 25 at 19:16
What the notation $(cdot ) ^+$ means?
– Cesareo
Jul 25 at 23:32
@Cesareo $(z)^+=z$ if $z>0$ and $(z)^+=0$ if $zleq 0$ in other words : $(z)^+=zmathbb I_z>0$
– the-owner
Jul 26 at 5:36