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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    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





















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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











      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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