Multi-Objective Linear Optimisation Pareto Solution
$begingroup$
I'm looking at optimising weightings over a set of different amounts to achieve a total, whilst minimising the amount over the total these are.
Think: I have macronutrient targets, I want to make sure I hit them but don't go over by an excessive amount so what 'weight' of each meal would I want.
So the constraint is:
$left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) ge left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)$
Whilst we want to minimise:
$mathrm{min} left[left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) - left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)right]$
I'm having a bit of trouble on where to start with this though. I know that for a problem with two objectives I could just make one objective as efficient as possible, but not really sure where to start here. And of course if the matrix P was 3x3 if it has an inverse the problem is automatically solved.
I'd also possibly want to expand this to more than three weightings. So any algorithm would need to be extendable to n-objectives and indeed m payoffs.
Are there any names of specific algorithms for this? Ideally something that is suitable for automation given a P matrix and a T vector.
optimization convex-optimization linear-programming
asked Jan 3 at 10:27
user403033user403033
357
$endgroup$
add a comment |
$begingroup$
I'm looking at optimising weightings over a set of different amounts to achieve a total, whilst minimising the amount over the total these are.
Think: I have macronutrient targets, I want to make sure I hit them but don't go over by an excessive amount so what 'weight' of each meal would I want.
So the constraint is:
$left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) ge left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)$
Whilst we want to minimise:
$mathrm{min} left[left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) - left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)right]$
I'm having a bit of trouble on where to start with this though. I know that for a problem with two objectives I could just make one objective as efficient as possible, but not really sure where to start here. And of course if the matrix P was 3x3 if it has an inverse the problem is automatically solved.
I'd also possibly want to expand this to more than three weightings. So any algorithm would need to be extendable to n-objectives and indeed m payoffs.
Are there any names of specific algorithms for this? Ideally something that is suitable for automation given a P matrix and a T vector.
optimization convex-optimization linear-programming
asked Jan 3 at 10:27
user403033user403033
357
$endgroup$
$begingroup$
are you just looking for any Pareto Solution?
$endgroup$
– LinAlg
Jan 3 at 12:44
$begingroup$
Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
$endgroup$
– user403033
Jan 3 at 14:40
$begingroup$
Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
$endgroup$
– callculus
Jan 3 at 19:05
$begingroup$
No, the weights would not have to sum to 1 but they would need to be positive.
$endgroup$
– user403033
Jan 4 at 9:45
add a comment |
$begingroup$
I'm looking at optimising weightings over a set of different amounts to achieve a total, whilst minimising the amount over the total these are.
Think: I have macronutrient targets, I want to make sure I hit them but don't go over by an excessive amount so what 'weight' of each meal would I want.
So the constraint is:
$left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) ge left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)$
Whilst we want to minimise:
$mathrm{min} left[left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) - left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)right]$
I'm having a bit of trouble on where to start with this though. I know that for a problem with two objectives I could just make one objective as efficient as possible, but not really sure where to start here. And of course if the matrix P was 3x3 if it has an inverse the problem is automatically solved.
I'd also possibly want to expand this to more than three weightings. So any algorithm would need to be extendable to n-objectives and indeed m payoffs.
Are there any names of specific algorithms for this? Ideally something that is suitable for automation given a P matrix and a T vector.
optimization convex-optimization linear-programming
asked Jan 3 at 10:27
user403033user403033
357
$endgroup$
I'm looking at optimising weightings over a set of different amounts to achieve a total, whilst minimising the amount over the total these are.
Think: I have macronutrient targets, I want to make sure I hit them but don't go over by an excessive amount so what 'weight' of each meal would I want.
So the constraint is:
$left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) ge left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)$
Whilst we want to minimise:
$mathrm{min} left[left( begin{array}
QP_{11} & P_{12} & P_{13} \
P_{21} & P_{22} & P_{23} \
P_{31} & P_{32} & P_{33} \
P_{41} & P_{42} & P_{43} end{array}right) cdot
left(begin{array} Qw_1 \
w_2 \
w_3 end{array} right) - left( begin{array} QT_1 \
T_2 \
T_3 \
T_4 end{array} right)right]$
I'm having a bit of trouble on where to start with this though. I know that for a problem with two objectives I could just make one objective as efficient as possible, but not really sure where to start here. And of course if the matrix P was 3x3 if it has an inverse the problem is automatically solved.
I'd also possibly want to expand this to more than three weightings. So any algorithm would need to be extendable to n-objectives and indeed m payoffs.
Are there any names of specific algorithms for this? Ideally something that is suitable for automation given a P matrix and a T vector.
optimization convex-optimization linear-programming
optimization convex-optimization linear-programming
asked Jan 3 at 10:27
user403033user403033
357
asked Jan 3 at 10:27
user403033user403033
357
asked Jan 3 at 10:27
user403033user403033
357
asked Jan 3 at 10:27
user403033user403033
357
asked Jan 3 at 10:27
user403033user403033
357
357
$begingroup$
are you just looking for any Pareto Solution?
$endgroup$
– LinAlg
Jan 3 at 12:44
$begingroup$
Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
$endgroup$
– user403033
Jan 3 at 14:40
$begingroup$
Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
$endgroup$
– callculus
Jan 3 at 19:05
$begingroup$
No, the weights would not have to sum to 1 but they would need to be positive.
$endgroup$
– user403033
Jan 4 at 9:45
add a comment |
$begingroup$
are you just looking for any Pareto Solution?
$endgroup$
– LinAlg
Jan 3 at 12:44
$begingroup$
Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
$endgroup$
– user403033
Jan 3 at 14:40
$begingroup$
Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
$endgroup$
– callculus
Jan 3 at 19:05
$begingroup$
No, the weights would not have to sum to 1 but they would need to be positive.
$endgroup$
– user403033
Jan 4 at 9:45
$begingroup$
are you just looking for any Pareto Solution?
$endgroup$
– LinAlg
Jan 3 at 12:44
$begingroup$
are you just looking for any Pareto Solution?
$endgroup$
– LinAlg
Jan 3 at 12:44
$begingroup$
Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
$endgroup$
– user403033
Jan 3 at 14:40
$begingroup$
Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
$endgroup$
– user403033
Jan 3 at 14:40
$begingroup$
Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
$endgroup$
– callculus
Jan 3 at 19:05
$begingroup$
Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
$endgroup$
– callculus
Jan 3 at 19:05
$begingroup$
No, the weights would not have to sum to 1 but they would need to be positive.
$endgroup$
– user403033
Jan 4 at 9:45
$begingroup$
No, the weights would not have to sum to 1 but they would need to be positive.
$endgroup$
– user403033
Jan 4 at 9:45
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your problem can be summarized as $min_{s,x}{s : Ax+sgeq b, sgeq 0}$. Weighted sum optimization gives a Pareto Solution as long as each weight is strictly positive:
$min_{x,s}{w^Ts : Ax+sgeq b, sgeq 0}$. These problems can be solved with the simplex method (available with linprog in scipy or Matlab).
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
$endgroup$
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your problem can be summarized as $min_{s,x}{s : Ax+sgeq b, sgeq 0}$. Weighted sum optimization gives a Pareto Solution as long as each weight is strictly positive:
$min_{x,s}{w^Ts : Ax+sgeq b, sgeq 0}$. These problems can be solved with the simplex method (available with linprog in scipy or Matlab).
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
$endgroup$
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
add a comment |
$begingroup$
Your problem can be summarized as $min_{s,x}{s : Ax+sgeq b, sgeq 0}$. Weighted sum optimization gives a Pareto Solution as long as each weight is strictly positive:
$min_{x,s}{w^Ts : Ax+sgeq b, sgeq 0}$. These problems can be solved with the simplex method (available with linprog in scipy or Matlab).
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
$endgroup$
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
add a comment |
$begingroup$
Your problem can be summarized as $min_{s,x}{s : Ax+sgeq b, sgeq 0}$. Weighted sum optimization gives a Pareto Solution as long as each weight is strictly positive:
$min_{x,s}{w^Ts : Ax+sgeq b, sgeq 0}$. These problems can be solved with the simplex method (available with linprog in scipy or Matlab).
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
$endgroup$
Your problem can be summarized as $min_{s,x}{s : Ax+sgeq b, sgeq 0}$. Weighted sum optimization gives a Pareto Solution as long as each weight is strictly positive:
$min_{x,s}{w^Ts : Ax+sgeq b, sgeq 0}$. These problems can be solved with the simplex method (available with linprog in scipy or Matlab).
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
answered Jan 4 at 2:12
LinAlgLinAlg
10.1k1521
10.1k1521
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
add a comment |
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
I assume in this instance $s$ is the vector of my minimisation condition and $w$ are the weights I'm applying to each? So that I'm then left with a scalar to minimise?
$endgroup$
– user403033
Jan 4 at 9:48
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
@user403033 that is correct!
$endgroup$
– LinAlg
Jan 4 at 14:20
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
Thanks, I'll give it a go with weightings of 1 across the board to start. A scalar condition will be much easier to deal with though. I assume if a particular entry of the P matrix is dominating the solution (due to relative size etc), to avoid this the problem is going to need to be reformulated?
$endgroup$
– user403033
Jan 4 at 15:03
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
$begingroup$
@user403033 you can either adjust the weights, or constrain two values of $P$ while optimizing the third value
$endgroup$
– LinAlg
Jan 16 at 20:06
add a comment |
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$begingroup$
are you just looking for any Pareto Solution?
$endgroup$
– LinAlg
Jan 3 at 12:44
$begingroup$
Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
$endgroup$
– user403033
Jan 3 at 14:40
$begingroup$
Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
$endgroup$
– callculus
Jan 3 at 19:05
$begingroup$
No, the weights would not have to sum to 1 but they would need to be positive.
$endgroup$
– user403033
Jan 4 at 9:45