Maximal value, linear programming problem The 2019 Stack Overflow Developer Survey Results Are InWhy maximum/minimum of linear programming occurs at a vertex?On the Proof of Fundamental Theorem of Linear Programming.Primal and dual solution to linear programmingLinear combination question in Linear Programming ProblemConverting linear programming problems into standard formMinimize the minimum - Linear programmingSingle nonzero value constraint formulation in linear programming problem statementEasier way of finding out whether a given linear programming problem has optimal solution or notSolving linear programming problem with given informationUtilizing theorems of duality to solve primal linear programming problemProof of 100 % rule in Linear ProgrammingLinear programming model for a retail online order
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Maximal value, linear programming problem
The 2019 Stack Overflow Developer Survey Results Are InWhy maximum/minimum of linear programming occurs at a vertex?On the Proof of Fundamental Theorem of Linear Programming.Primal and dual solution to linear programmingLinear combination question in Linear Programming ProblemConverting linear programming problems into standard formMinimize the minimum - Linear programmingSingle nonzero value constraint formulation in linear programming problem statementEasier way of finding out whether a given linear programming problem has optimal solution or notSolving linear programming problem with given informationUtilizing theorems of duality to solve primal linear programming problemProof of 100 % rule in Linear ProgrammingLinear programming model for a retail online order
$begingroup$
I want to find the maximal $psi_1$ for the following linear programming problem:
beginalign
max frac23025(525+121psi_3 + 1089 psi_4), text s.t.
\
endalign
beginarray
text0 leq 1450 - 3267 psi_3 - 5203 psi_4 \
text0 leq psi_3\
text0 leq psi_4
endarray
Intuitively we want to make $psi_3$ and $psi_4$ as large as possible.
My textbook states that:
Since the constraints are linear, it follows that either all the
weight should be put on $psi_3$ or $psi_4$.
Why is that the case?
I have drawn the constraints, I see that the possible values should be inside a triangle. But why the optimal values should be one of the two corners of the triangle?
optimization linear-programming
asked Apr 6 at 20:38
VictorVictor
577
$endgroup$
add a comment |
$begingroup$
I want to find the maximal $psi_1$ for the following linear programming problem:
beginalign
max frac23025(525+121psi_3 + 1089 psi_4), text s.t.
\
endalign
beginarray
text0 leq 1450 - 3267 psi_3 - 5203 psi_4 \
text0 leq psi_3\
text0 leq psi_4
endarray
Intuitively we want to make $psi_3$ and $psi_4$ as large as possible.
My textbook states that:
Since the constraints are linear, it follows that either all the
weight should be put on $psi_3$ or $psi_4$.
Why is that the case?
I have drawn the constraints, I see that the possible values should be inside a triangle. But why the optimal values should be one of the two corners of the triangle?
optimization linear-programming
asked Apr 6 at 20:38
VictorVictor
577
$endgroup$
1
$begingroup$
In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region.
$endgroup$
– Brian Borchers
Apr 6 at 20:43
$begingroup$
@BrianBorchers Do you have a proof or some kind of intuition of why this is the case?
$endgroup$
– Victor
Apr 6 at 20:48
2
$begingroup$
This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method.
$endgroup$
– Brian Borchers
Apr 6 at 21:08
$begingroup$
This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here.
$endgroup$
– David M.
Apr 7 at 18:26
add a comment |
$begingroup$
I want to find the maximal $psi_1$ for the following linear programming problem:
beginalign
max frac23025(525+121psi_3 + 1089 psi_4), text s.t.
\
endalign
beginarray
text0 leq 1450 - 3267 psi_3 - 5203 psi_4 \
text0 leq psi_3\
text0 leq psi_4
endarray
Intuitively we want to make $psi_3$ and $psi_4$ as large as possible.
My textbook states that:
Since the constraints are linear, it follows that either all the
weight should be put on $psi_3$ or $psi_4$.
Why is that the case?
I have drawn the constraints, I see that the possible values should be inside a triangle. But why the optimal values should be one of the two corners of the triangle?
optimization linear-programming
asked Apr 6 at 20:38
VictorVictor
577
$endgroup$
I want to find the maximal $psi_1$ for the following linear programming problem:
beginalign
max frac23025(525+121psi_3 + 1089 psi_4), text s.t.
\
endalign
beginarray
text0 leq 1450 - 3267 psi_3 - 5203 psi_4 \
text0 leq psi_3\
text0 leq psi_4
endarray
Intuitively we want to make $psi_3$ and $psi_4$ as large as possible.
My textbook states that:
Since the constraints are linear, it follows that either all the
weight should be put on $psi_3$ or $psi_4$.
Why is that the case?
I have drawn the constraints, I see that the possible values should be inside a triangle. But why the optimal values should be one of the two corners of the triangle?
optimization linear-programming
optimization linear-programming
asked Apr 6 at 20:38
VictorVictor
577
asked Apr 6 at 20:38
VictorVictor
577
asked Apr 6 at 20:38
VictorVictor
577
asked Apr 6 at 20:38
VictorVictor
577
asked Apr 6 at 20:38
VictorVictor
577
577
1
$begingroup$
In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region.
$endgroup$
– Brian Borchers
Apr 6 at 20:43
$begingroup$
@BrianBorchers Do you have a proof or some kind of intuition of why this is the case?
$endgroup$
– Victor
Apr 6 at 20:48
2
$begingroup$
This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method.
$endgroup$
– Brian Borchers
Apr 6 at 21:08
$begingroup$
This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here.
$endgroup$
– David M.
Apr 7 at 18:26
add a comment |
1
$begingroup$
In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region.
$endgroup$
– Brian Borchers
Apr 6 at 20:43
$begingroup$
@BrianBorchers Do you have a proof or some kind of intuition of why this is the case?
$endgroup$
– Victor
Apr 6 at 20:48
2
$begingroup$
This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method.
$endgroup$
– Brian Borchers
Apr 6 at 21:08
$begingroup$
This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here.
$endgroup$
– David M.
Apr 7 at 18:26
1
1
$begingroup$
In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region.
$endgroup$
– Brian Borchers
Apr 6 at 20:43
$begingroup$
In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region.
$endgroup$
– Brian Borchers
Apr 6 at 20:43
$begingroup$
@BrianBorchers Do you have a proof or some kind of intuition of why this is the case?
$endgroup$
– Victor
Apr 6 at 20:48
$begingroup$
@BrianBorchers Do you have a proof or some kind of intuition of why this is the case?
$endgroup$
– Victor
Apr 6 at 20:48
2
2
$begingroup$
This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method.
$endgroup$
– Brian Borchers
Apr 6 at 21:08
$begingroup$
This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method.
$endgroup$
– Brian Borchers
Apr 6 at 21:08
$begingroup$
This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here.
$endgroup$
– David M.
Apr 7 at 18:26
$begingroup$
This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here.
$endgroup$
– David M.
Apr 7 at 18:26
add a comment |
0
active
oldest
votes
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1
$begingroup$
In a linear program, it's always the case that if there is an optimal solution (the LP isn't unbounded), then there is an optimal solution that lies at a corner of the feasible region. There may be additional optimal solutions (with the same optimal objective value) along edges or higher dimensional faces of the feasible region.
$endgroup$
– Brian Borchers
Apr 6 at 20:43
$begingroup$
@BrianBorchers Do you have a proof or some kind of intuition of why this is the case?
$endgroup$
– Victor
Apr 6 at 20:48
2
$begingroup$
This is a standard theorem in linear programming that is usually proved in courses on that subject. Chvatal's Linear Programming has a nice constructive proof based on the simplex method.
$endgroup$
– Brian Borchers
Apr 6 at 21:08
$begingroup$
This topic (called the fundamental theorem of linear programming), is discussed here with some follow-up here.
$endgroup$
– David M.
Apr 7 at 18:26