How to explain that the the following points are optimal in a region?
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I have a region defined by following inequalities $$yleq min(a,b)tag1$$ $$xleqfraccdc+d-1tag2$$ $$fracc+d-1cx+fracdmin(a,e)yleq dtag3$$where $a,b,c,d,e$ are positive constants. In this case how can we show that if $$min(a,b)<min(a,e)\min(a,e)leq fraccdc+d-1$$ then the region associated with (1)-(3) is given by the following Figure (1) where the optimal point is shown by the circle (the optimal point is the point which maximizes the left hand side of (3))
On the other hand if $$min(a,b)<min(a,e)\ min(a,e)>fraccdc+d-1$$ then the associated region and the optimal point are given by the following figure 
I do not understand how the optimal point in second case is as shown in the Figure. Any explanation in this regard will be much appreciated. Thanks in advance.
Here is the image that I saw in a research paper. In this figure we have $$a=K_u,b=M_R,c=N_d,d=K_d,e=N_b$$. The optimal points are mentioned with dark circles on the graph.
real-analysis optimization real-numbers
asked Sep 10 at 7:21
Frank Moses
1,129317
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up vote
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I have a region defined by following inequalities $$yleq min(a,b)tag1$$ $$xleqfraccdc+d-1tag2$$ $$fracc+d-1cx+fracdmin(a,e)yleq dtag3$$where $a,b,c,d,e$ are positive constants. In this case how can we show that if $$min(a,b)<min(a,e)\min(a,e)leq fraccdc+d-1$$ then the region associated with (1)-(3) is given by the following Figure (1) where the optimal point is shown by the circle (the optimal point is the point which maximizes the left hand side of (3))
On the other hand if $$min(a,b)<min(a,e)\ min(a,e)>fraccdc+d-1$$ then the associated region and the optimal point are given by the following figure
I do not understand how the optimal point in second case is as shown in the Figure. Any explanation in this regard will be much appreciated. Thanks in advance.
Here is the image that I saw in a research paper. In this figure we have $$a=K_u,b=M_R,c=N_d,d=K_d,e=N_b$$. The optimal points are mentioned with dark circles on the graph.
real-analysis optimization real-numbers
asked Sep 10 at 7:21
Frank Moses
1,129317
1
Isn't the objective value on the entire diagonal line equal to $d$? Or is the sketch misleading?
– LinAlg
Sep 10 at 13:06
@LinAlg I also think the same but then I do not know why the circled point are considered to be the optimal points in the research paper that I read. You can see the figure from the paper that I have attached into my post.
– Frank Moses
Sep 11 at 0:01
Isn't the clou in the description of Fig. 4 at the left bottom of page 7202?
– LinAlg
Sep 11 at 0:28
@LinAlg Actually I do not see any clou and I do not understand that clou. I can understand the first three parts of the Figure but I am really unable to understand the placement of the dark circle in last part of the figure. I will be very thankful to you if you elaborate what that clou is. Thank you so much for your interest. (BTW I am extremely surprised that you got to the exact paper without any clou about the paper from the original post. Is your research area wireless communication?)
– Frank Moses
Sep 11 at 1:02
@LinAlg oh I see now. The optimal point is defined as the maximum possible sum of $LDoF_u$ and $LDoF_d$.
– Frank Moses
Sep 11 at 5:29
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a region defined by following inequalities $$yleq min(a,b)tag1$$ $$xleqfraccdc+d-1tag2$$ $$fracc+d-1cx+fracdmin(a,e)yleq dtag3$$where $a,b,c,d,e$ are positive constants. In this case how can we show that if $$min(a,b)<min(a,e)\min(a,e)leq fraccdc+d-1$$ then the region associated with (1)-(3) is given by the following Figure (1) where the optimal point is shown by the circle (the optimal point is the point which maximizes the left hand side of (3))
On the other hand if $$min(a,b)<min(a,e)\ min(a,e)>fraccdc+d-1$$ then the associated region and the optimal point are given by the following figure
I do not understand how the optimal point in second case is as shown in the Figure. Any explanation in this regard will be much appreciated. Thanks in advance.
Here is the image that I saw in a research paper. In this figure we have $$a=K_u,b=M_R,c=N_d,d=K_d,e=N_b$$. The optimal points are mentioned with dark circles on the graph.
real-analysis optimization real-numbers
asked Sep 10 at 7:21
Frank Moses
1,129317
I have a region defined by following inequalities $$yleq min(a,b)tag1$$ $$xleqfraccdc+d-1tag2$$ $$fracc+d-1cx+fracdmin(a,e)yleq dtag3$$where $a,b,c,d,e$ are positive constants. In this case how can we show that if $$min(a,b)<min(a,e)\min(a,e)leq fraccdc+d-1$$ then the region associated with (1)-(3) is given by the following Figure (1) where the optimal point is shown by the circle (the optimal point is the point which maximizes the left hand side of (3))
On the other hand if $$min(a,b)<min(a,e)\ min(a,e)>fraccdc+d-1$$ then the associated region and the optimal point are given by the following figure
I do not understand how the optimal point in second case is as shown in the Figure. Any explanation in this regard will be much appreciated. Thanks in advance.
Here is the image that I saw in a research paper. In this figure we have $$a=K_u,b=M_R,c=N_d,d=K_d,e=N_b$$. The optimal points are mentioned with dark circles on the graph.
real-analysis optimization real-numbers
real-analysis optimization real-numbers
asked Sep 10 at 7:21
Frank Moses
1,129317
asked Sep 10 at 7:21
Frank Moses
1,129317
edited Sep 11 at 0:05
asked Sep 10 at 7:21
Frank Moses
1,129317
asked Sep 10 at 7:21
Frank Moses
1,129317
asked Sep 10 at 7:21
Frank Moses
1,129317
1,129317
1
Isn't the objective value on the entire diagonal line equal to $d$? Or is the sketch misleading?
– LinAlg
Sep 10 at 13:06
@LinAlg I also think the same but then I do not know why the circled point are considered to be the optimal points in the research paper that I read. You can see the figure from the paper that I have attached into my post.
– Frank Moses
Sep 11 at 0:01
Isn't the clou in the description of Fig. 4 at the left bottom of page 7202?
– LinAlg
Sep 11 at 0:28
@LinAlg Actually I do not see any clou and I do not understand that clou. I can understand the first three parts of the Figure but I am really unable to understand the placement of the dark circle in last part of the figure. I will be very thankful to you if you elaborate what that clou is. Thank you so much for your interest. (BTW I am extremely surprised that you got to the exact paper without any clou about the paper from the original post. Is your research area wireless communication?)
– Frank Moses
Sep 11 at 1:02
@LinAlg oh I see now. The optimal point is defined as the maximum possible sum of $LDoF_u$ and $LDoF_d$.
– Frank Moses
Sep 11 at 5:29
 |Â
show 1 more comment
1
Isn't the objective value on the entire diagonal line equal to $d$? Or is the sketch misleading?
– LinAlg
Sep 10 at 13:06
@LinAlg I also think the same but then I do not know why the circled point are considered to be the optimal points in the research paper that I read. You can see the figure from the paper that I have attached into my post.
– Frank Moses
Sep 11 at 0:01
Isn't the clou in the description of Fig. 4 at the left bottom of page 7202?
– LinAlg
Sep 11 at 0:28
@LinAlg Actually I do not see any clou and I do not understand that clou. I can understand the first three parts of the Figure but I am really unable to understand the placement of the dark circle in last part of the figure. I will be very thankful to you if you elaborate what that clou is. Thank you so much for your interest. (BTW I am extremely surprised that you got to the exact paper without any clou about the paper from the original post. Is your research area wireless communication?)
– Frank Moses
Sep 11 at 1:02
@LinAlg oh I see now. The optimal point is defined as the maximum possible sum of $LDoF_u$ and $LDoF_d$.
– Frank Moses
Sep 11 at 5:29
1
1
Isn't the objective value on the entire diagonal line equal to $d$? Or is the sketch misleading?
– LinAlg
Sep 10 at 13:06
Isn't the objective value on the entire diagonal line equal to $d$? Or is the sketch misleading?
– LinAlg
Sep 10 at 13:06
@LinAlg I also think the same but then I do not know why the circled point are considered to be the optimal points in the research paper that I read. You can see the figure from the paper that I have attached into my post.
– Frank Moses
Sep 11 at 0:01
@LinAlg I also think the same but then I do not know why the circled point are considered to be the optimal points in the research paper that I read. You can see the figure from the paper that I have attached into my post.
– Frank Moses
Sep 11 at 0:01
Isn't the clou in the description of Fig. 4 at the left bottom of page 7202?
– LinAlg
Sep 11 at 0:28
Isn't the clou in the description of Fig. 4 at the left bottom of page 7202?
– LinAlg
Sep 11 at 0:28
@LinAlg Actually I do not see any clou and I do not understand that clou. I can understand the first three parts of the Figure but I am really unable to understand the placement of the dark circle in last part of the figure. I will be very thankful to you if you elaborate what that clou is. Thank you so much for your interest. (BTW I am extremely surprised that you got to the exact paper without any clou about the paper from the original post. Is your research area wireless communication?)
– Frank Moses
Sep 11 at 1:02
@LinAlg Actually I do not see any clou and I do not understand that clou. I can understand the first three parts of the Figure but I am really unable to understand the placement of the dark circle in last part of the figure. I will be very thankful to you if you elaborate what that clou is. Thank you so much for your interest. (BTW I am extremely surprised that you got to the exact paper without any clou about the paper from the original post. Is your research area wireless communication?)
– Frank Moses
Sep 11 at 1:02
@LinAlg oh I see now. The optimal point is defined as the maximum possible sum of $LDoF_u$ and $LDoF_d$.
– Frank Moses
Sep 11 at 5:29
@LinAlg oh I see now. The optimal point is defined as the maximum possible sum of $LDoF_u$ and $LDoF_d$.
– Frank Moses
Sep 11 at 5:29
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
In the paper (found via Google Scholar), the objective value is LDoF$_sum$ (see either the description or the caption of Fig. 4), defined in formula (6) as LDoF$_u$+LDoF$_d$. So, the optimal point is not the one that maximizes (3).
Which point is optimal is determined by the slope of the diagonal line. If it is steeper than 45 degrees, the bottom point is optimal. If it is more horizontal, the top corner point is optimal.
For fig. (4d), the two conditions under the figure oppose each other: the first one makes the line more horizontal, while the second one makes the line less horizontal. This case is therefore not as simple as the other ones. Let's check the objective value at the circled point:
$$beginalign
LDoF_u + LDoF_d &= min(K_u,M_r) + fracN_dN_d+K_d-1 left(K_d - fracK_dmin(K_u,N_b) min(K_u,M_r)right)\
&= fracN_dK_dN_d+K_d-1 + min(K_u,M_r) - fracN_dK_dN_d+K_d-1 fracmin(K_u,M_r)min(K_u,N_b)\
endalign$$
The second term on the first line is obtained by solving (23) for LDoF$_d$. This equals the right hand side of the inequality at the left bottom of page 7202, which according to Theorem 1 is optimal.
answered Sep 11 at 13:19
LinAlg
6,0591319
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
In the paper (found via Google Scholar), the objective value is LDoF$_sum$ (see either the description or the caption of Fig. 4), defined in formula (6) as LDoF$_u$+LDoF$_d$. So, the optimal point is not the one that maximizes (3).
Which point is optimal is determined by the slope of the diagonal line. If it is steeper than 45 degrees, the bottom point is optimal. If it is more horizontal, the top corner point is optimal.
For fig. (4d), the two conditions under the figure oppose each other: the first one makes the line more horizontal, while the second one makes the line less horizontal. This case is therefore not as simple as the other ones. Let's check the objective value at the circled point:
$$beginalign
LDoF_u + LDoF_d &= min(K_u,M_r) + fracN_dN_d+K_d-1 left(K_d - fracK_dmin(K_u,N_b) min(K_u,M_r)right)\
&= fracN_dK_dN_d+K_d-1 + min(K_u,M_r) - fracN_dK_dN_d+K_d-1 fracmin(K_u,M_r)min(K_u,N_b)\
endalign$$
The second term on the first line is obtained by solving (23) for LDoF$_d$. This equals the right hand side of the inequality at the left bottom of page 7202, which according to Theorem 1 is optimal.
answered Sep 11 at 13:19
LinAlg
6,0591319
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
add a comment |Â
up vote
1
down vote
accepted
In the paper (found via Google Scholar), the objective value is LDoF$_sum$ (see either the description or the caption of Fig. 4), defined in formula (6) as LDoF$_u$+LDoF$_d$. So, the optimal point is not the one that maximizes (3).
Which point is optimal is determined by the slope of the diagonal line. If it is steeper than 45 degrees, the bottom point is optimal. If it is more horizontal, the top corner point is optimal.
For fig. (4d), the two conditions under the figure oppose each other: the first one makes the line more horizontal, while the second one makes the line less horizontal. This case is therefore not as simple as the other ones. Let's check the objective value at the circled point:
$$beginalign
LDoF_u + LDoF_d &= min(K_u,M_r) + fracN_dN_d+K_d-1 left(K_d - fracK_dmin(K_u,N_b) min(K_u,M_r)right)\
&= fracN_dK_dN_d+K_d-1 + min(K_u,M_r) - fracN_dK_dN_d+K_d-1 fracmin(K_u,M_r)min(K_u,N_b)\
endalign$$
The second term on the first line is obtained by solving (23) for LDoF$_d$. This equals the right hand side of the inequality at the left bottom of page 7202, which according to Theorem 1 is optimal.
answered Sep 11 at 13:19
LinAlg
6,0591319
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
In the paper (found via Google Scholar), the objective value is LDoF$_sum$ (see either the description or the caption of Fig. 4), defined in formula (6) as LDoF$_u$+LDoF$_d$. So, the optimal point is not the one that maximizes (3).
Which point is optimal is determined by the slope of the diagonal line. If it is steeper than 45 degrees, the bottom point is optimal. If it is more horizontal, the top corner point is optimal.
For fig. (4d), the two conditions under the figure oppose each other: the first one makes the line more horizontal, while the second one makes the line less horizontal. This case is therefore not as simple as the other ones. Let's check the objective value at the circled point:
$$beginalign
LDoF_u + LDoF_d &= min(K_u,M_r) + fracN_dN_d+K_d-1 left(K_d - fracK_dmin(K_u,N_b) min(K_u,M_r)right)\
&= fracN_dK_dN_d+K_d-1 + min(K_u,M_r) - fracN_dK_dN_d+K_d-1 fracmin(K_u,M_r)min(K_u,N_b)\
endalign$$
The second term on the first line is obtained by solving (23) for LDoF$_d$. This equals the right hand side of the inequality at the left bottom of page 7202, which according to Theorem 1 is optimal.
answered Sep 11 at 13:19
LinAlg
6,0591319
In the paper (found via Google Scholar), the objective value is LDoF$_sum$ (see either the description or the caption of Fig. 4), defined in formula (6) as LDoF$_u$+LDoF$_d$. So, the optimal point is not the one that maximizes (3).
Which point is optimal is determined by the slope of the diagonal line. If it is steeper than 45 degrees, the bottom point is optimal. If it is more horizontal, the top corner point is optimal.
For fig. (4d), the two conditions under the figure oppose each other: the first one makes the line more horizontal, while the second one makes the line less horizontal. This case is therefore not as simple as the other ones. Let's check the objective value at the circled point:
$$beginalign
LDoF_u + LDoF_d &= min(K_u,M_r) + fracN_dN_d+K_d-1 left(K_d - fracK_dmin(K_u,N_b) min(K_u,M_r)right)\
&= fracN_dK_dN_d+K_d-1 + min(K_u,M_r) - fracN_dK_dN_d+K_d-1 fracmin(K_u,M_r)min(K_u,N_b)\
endalign$$
The second term on the first line is obtained by solving (23) for LDoF$_d$. This equals the right hand side of the inequality at the left bottom of page 7202, which according to Theorem 1 is optimal.
answered Sep 11 at 13:19
LinAlg
6,0591319
answered Sep 11 at 13:19
LinAlg
6,0591319
answered Sep 11 at 13:19
LinAlg
6,0591319
answered Sep 11 at 13:19
LinAlg
6,0591319
6,0591319
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
add a comment |Â
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
Thank you so much for detailed explanation.
– Frank Moses
Sep 12 at 4:28
add a comment |Â
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1
Isn't the objective value on the entire diagonal line equal to $d$? Or is the sketch misleading?
– LinAlg
Sep 10 at 13:06
@LinAlg I also think the same but then I do not know why the circled point are considered to be the optimal points in the research paper that I read. You can see the figure from the paper that I have attached into my post.
– Frank Moses
Sep 11 at 0:01
Isn't the clou in the description of Fig. 4 at the left bottom of page 7202?
– LinAlg
Sep 11 at 0:28
@LinAlg Actually I do not see any clou and I do not understand that clou. I can understand the first three parts of the Figure but I am really unable to understand the placement of the dark circle in last part of the figure. I will be very thankful to you if you elaborate what that clou is. Thank you so much for your interest. (BTW I am extremely surprised that you got to the exact paper without any clou about the paper from the original post. Is your research area wireless communication?)
– Frank Moses
Sep 11 at 1:02
@LinAlg oh I see now. The optimal point is defined as the maximum possible sum of $LDoF_u$ and $LDoF_d$.
– Frank Moses
Sep 11 at 5:29