Solutions of two linear programming

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Let $betaequiv (beta_0, beta_1)in mathcalBsubset mathbbR^2$ with $mathcalB$ compact. $beta$ is a known vector of parameters.

Let $P_0, P_1, F_0, F_1$ be know parameters, each in $[0,1]$.

Consider the following linear programming problems.

beginaligned
underlinep(beta)equiv min_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

and

beginaligned
barp(beta)equiv max_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

---

Question: Is it true that $forall p in [underlinep(beta),barp(beta)]$, there exists $tildebetain mathcalB$ such that

beginaligned
& 1) text [P_0-a]F_0+ [b - P_1]F_1 = p text has a solution wrto $a,b$\
& 2) text ain [0,1]text, bin [0,1]\
& 3) text Q_0: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(tildebeta_0) = P_0 text, Q_0( tildebeta_0+tildebeta_1)= a\
& text is weakly increasing\
& 4) text Q_1: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(tildebeta_0) = b text, Q_1( tildebeta_0+tildebeta_1)= P_1\
&text is weakly increasing
endaligned
? Why yes or not? I think the answer should be yes, but your hint would be very appreciated.

linear-algebra optimization linear-programming maxima-minima

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asked Jul 23 at 21:56

TEX

2419

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up vote
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Let $betaequiv (beta_0, beta_1)in mathcalBsubset mathbbR^2$ with $mathcalB$ compact. $beta$ is a known vector of parameters.

Let $P_0, P_1, F_0, F_1$ be know parameters, each in $[0,1]$.

Consider the following linear programming problems.

beginaligned
underlinep(beta)equiv min_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

and

beginaligned
barp(beta)equiv max_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

---

Question: Is it true that $forall p in [underlinep(beta),barp(beta)]$, there exists $tildebetain mathcalB$ such that

beginaligned
& 1) text [P_0-a]F_0+ [b - P_1]F_1 = p text has a solution wrto $a,b$\
& 2) text ain [0,1]text, bin [0,1]\
& 3) text Q_0: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(tildebeta_0) = P_0 text, Q_0( tildebeta_0+tildebeta_1)= a\
& text is weakly increasing\
& 4) text Q_1: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(tildebeta_0) = b text, Q_1( tildebeta_0+tildebeta_1)= P_1\
&text is weakly increasing
endaligned
? Why yes or not? I think the answer should be yes, but your hint would be very appreciated.

linear-algebra optimization linear-programming maxima-minima

share|cite|improve this question

asked Jul 23 at 21:56

TEX

2419

add a comment | 

up vote
2
down vote

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1

up vote
2
down vote

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1

Let $betaequiv (beta_0, beta_1)in mathcalBsubset mathbbR^2$ with $mathcalB$ compact. $beta$ is a known vector of parameters.

Let $P_0, P_1, F_0, F_1$ be know parameters, each in $[0,1]$.

Consider the following linear programming problems.

beginaligned
underlinep(beta)equiv min_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

and

beginaligned
barp(beta)equiv max_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

---

Question: Is it true that $forall p in [underlinep(beta),barp(beta)]$, there exists $tildebetain mathcalB$ such that

beginaligned
& 1) text [P_0-a]F_0+ [b - P_1]F_1 = p text has a solution wrto $a,b$\
& 2) text ain [0,1]text, bin [0,1]\
& 3) text Q_0: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(tildebeta_0) = P_0 text, Q_0( tildebeta_0+tildebeta_1)= a\
& text is weakly increasing\
& 4) text Q_1: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(tildebeta_0) = b text, Q_1( tildebeta_0+tildebeta_1)= P_1\
&text is weakly increasing
endaligned
? Why yes or not? I think the answer should be yes, but your hint would be very appreciated.

linear-algebra optimization linear-programming maxima-minima

share|cite|improve this question

asked Jul 23 at 21:56

TEX

2419

Let $betaequiv (beta_0, beta_1)in mathcalBsubset mathbbR^2$ with $mathcalB$ compact. $beta$ is a known vector of parameters.

Let $P_0, P_1, F_0, F_1$ be know parameters, each in $[0,1]$.

Consider the following linear programming problems.

beginaligned
underlinep(beta)equiv min_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

and

beginaligned
barp(beta)equiv max_a,b & [P_0-a]F_0+ [b - P_1]F_1 \
&text s.t. \
& 1) text ain [0,1]text, bin [0,1]\
& 2) text Q_0: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(beta_0) = P_0 text, Q_0( beta_0+beta_1)= a\
& text is weakly increasing\
& 3) text Q_1: 0, beta_0, beta_0+beta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(beta_0) = b text, Q_1( beta_0+beta_1)= P_1\
&text is weakly increasing
endaligned

---

Question: Is it true that $forall p in [underlinep(beta),barp(beta)]$, there exists $tildebetain mathcalB$ such that

beginaligned
& 1) text [P_0-a]F_0+ [b - P_1]F_1 = p text has a solution wrto $a,b$\
& 2) text ain [0,1]text, bin [0,1]\
& 3) text Q_0: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_0(0)=frac12text, Q_0(tildebeta_0) = P_0 text, Q_0( tildebeta_0+tildebeta_1)= a\
& text is weakly increasing\
& 4) text Q_1: 0, tildebeta_0, tildebeta_0+tildebeta_1 rightarrow [0,1] text with Q_1(0)=frac12text, Q_1(tildebeta_0) = b text, Q_1( tildebeta_0+tildebeta_1)= P_1\
&text is weakly increasing
endaligned
? Why yes or not? I think the answer should be yes, but your hint would be very appreciated.

linear-algebra optimization linear-programming maxima-minima

share|cite|improve this question

asked Jul 23 at 21:56

TEX

2419

share|cite|improve this question

share|cite|improve this question

asked Jul 23 at 21:56

TEX

2419

asked Jul 23 at 21:56

TEX

2419

asked Jul 23 at 21:56

TEX

2419

2419

add a comment | 

add a comment | 

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