Multi-Objective Linear Optimisation Pareto Solution












0














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







share|cite|improve this question






















  • are you just looking for any Pareto Solution?
    – LinAlg
    23 hours ago










  • Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
    – user403033
    22 hours ago










  • Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
    – callculus
    17 hours ago












  • No, the weights would not have to sum to 1 but they would need to be positive.
    – user403033
    2 hours ago
















0














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







share|cite|improve this question






















  • are you just looking for any Pareto Solution?
    – LinAlg
    23 hours ago










  • Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
    – user403033
    22 hours ago










  • Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
    – callculus
    17 hours ago












  • No, the weights would not have to sum to 1 but they would need to be positive.
    – user403033
    2 hours ago














0












0








0







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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked yesterday









user403033

256




256












  • are you just looking for any Pareto Solution?
    – LinAlg
    23 hours ago










  • Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
    – user403033
    22 hours ago










  • Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
    – callculus
    17 hours ago












  • No, the weights would not have to sum to 1 but they would need to be positive.
    – user403033
    2 hours ago


















  • are you just looking for any Pareto Solution?
    – LinAlg
    23 hours ago










  • Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
    – user403033
    22 hours ago










  • Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
    – callculus
    17 hours ago












  • No, the weights would not have to sum to 1 but they would need to be positive.
    – user403033
    2 hours ago
















are you just looking for any Pareto Solution?
– LinAlg
23 hours ago




are you just looking for any Pareto Solution?
– LinAlg
23 hours ago












Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
– user403033
22 hours ago




Any efficient solution would suffice, from there it should be possible to extend to all efficient solutions.
– user403033
22 hours ago












Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
– callculus
17 hours ago






Is $w_1+w_2+w_3=1$ and addtionally $w_igeq 0 forall i in {1,2,3 } $?
– callculus
17 hours ago














No, the weights would not have to sum to 1 but they would need to be positive.
– user403033
2 hours ago




No, the weights would not have to sum to 1 but they would need to be positive.
– user403033
2 hours ago










1 Answer
1






active

oldest

votes


















0














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).






share|cite|improve this answer





















  • 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?
    – user403033
    2 hours ago











Your Answer





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1 Answer
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active

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1 Answer
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active

oldest

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active

oldest

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active

oldest

votes









0














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).






share|cite|improve this answer





















  • 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?
    – user403033
    2 hours ago
















0














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).






share|cite|improve this answer





















  • 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?
    – user403033
    2 hours ago














0












0








0






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).






share|cite|improve this answer












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).







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 10 hours ago









LinAlg

8,4161521




8,4161521












  • 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?
    – user403033
    2 hours ago


















  • 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?
    – user403033
    2 hours ago
















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?
– user403033
2 hours ago




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?
– user403033
2 hours ago


















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