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Basic Feasible Solutions, Basic Solutions and Optimal Solution



The 2019 Stack Overflow Developer Survey Results Are InAre these solutions to a LP problem feasible? basic?Finding all basic feasible solutions in a linear programFind all optimal solutions by SimplexHow to test if a feasible solution is optimal - Complementary Slackness Theorem - Linear ProgrammingSimplex method - multiple optimal solutions?How to find all basic feasible solutions of a linear system?Primal-Dual basic (feasible) solution?Problem regarding a LPP can have a non-basic optimal solutionBasic Feasible solution And Optimality condition










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$begingroup$


I'm currently studying linear programming and I came across this MIT resource. In the very first page of the pdf, under BT Exercise 2.10 the 6th statement reads:




  1. Consider the problem of minimizing maxc’x, d’x over the set P. If
    this problem has an optimal solution, it must have an optimal solution
    which is an extreme point of P.



And on the next page, the solution manual author claims that the above statement is false and proceeds to provide a counter-example:




  1. False. Consider the problem of minimizing $x_1−0.5 = maxx_1−0.5x_3, −x_1+ 0.5x_3$ subject to $x_1+x_2 = 1, x_3 = 1$ and $x_1 + x_2 = 1, x_3 = 1$ and $(x_1,x_2,x_3) ≥ (0, 0, 0)$. Its unique optimal solution is $(x_1, x_2, x_3) =
    > (0.5, 0.5, 1)$
    which is not a BFS.



This confuses me. I was under the impression that extreme points are the same as basic solutions. Is this true?



If it is true and if one of basic solutions is not the optimal solution, then what is even the point of calculating the basic solutions? What am I missing here? Any help to get out of this confusion is appreciated!










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    I'm currently studying linear programming and I came across this MIT resource. In the very first page of the pdf, under BT Exercise 2.10 the 6th statement reads:




    1. Consider the problem of minimizing maxc’x, d’x over the set P. If
      this problem has an optimal solution, it must have an optimal solution
      which is an extreme point of P.



    And on the next page, the solution manual author claims that the above statement is false and proceeds to provide a counter-example:




    1. False. Consider the problem of minimizing $x_1−0.5 = maxx_1−0.5x_3, −x_1+ 0.5x_3$ subject to $x_1+x_2 = 1, x_3 = 1$ and $x_1 + x_2 = 1, x_3 = 1$ and $(x_1,x_2,x_3) ≥ (0, 0, 0)$. Its unique optimal solution is $(x_1, x_2, x_3) =
      > (0.5, 0.5, 1)$
      which is not a BFS.



    This confuses me. I was under the impression that extreme points are the same as basic solutions. Is this true?



    If it is true and if one of basic solutions is not the optimal solution, then what is even the point of calculating the basic solutions? What am I missing here? Any help to get out of this confusion is appreciated!










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      I'm currently studying linear programming and I came across this MIT resource. In the very first page of the pdf, under BT Exercise 2.10 the 6th statement reads:




      1. Consider the problem of minimizing maxc’x, d’x over the set P. If
        this problem has an optimal solution, it must have an optimal solution
        which is an extreme point of P.



      And on the next page, the solution manual author claims that the above statement is false and proceeds to provide a counter-example:




      1. False. Consider the problem of minimizing $x_1−0.5 = maxx_1−0.5x_3, −x_1+ 0.5x_3$ subject to $x_1+x_2 = 1, x_3 = 1$ and $x_1 + x_2 = 1, x_3 = 1$ and $(x_1,x_2,x_3) ≥ (0, 0, 0)$. Its unique optimal solution is $(x_1, x_2, x_3) =
        > (0.5, 0.5, 1)$
        which is not a BFS.



      This confuses me. I was under the impression that extreme points are the same as basic solutions. Is this true?



      If it is true and if one of basic solutions is not the optimal solution, then what is even the point of calculating the basic solutions? What am I missing here? Any help to get out of this confusion is appreciated!










      share|cite|improve this question











      $endgroup$




      I'm currently studying linear programming and I came across this MIT resource. In the very first page of the pdf, under BT Exercise 2.10 the 6th statement reads:




      1. Consider the problem of minimizing maxc’x, d’x over the set P. If
        this problem has an optimal solution, it must have an optimal solution
        which is an extreme point of P.



      And on the next page, the solution manual author claims that the above statement is false and proceeds to provide a counter-example:




      1. False. Consider the problem of minimizing $x_1−0.5 = maxx_1−0.5x_3, −x_1+ 0.5x_3$ subject to $x_1+x_2 = 1, x_3 = 1$ and $x_1 + x_2 = 1, x_3 = 1$ and $(x_1,x_2,x_3) ≥ (0, 0, 0)$. Its unique optimal solution is $(x_1, x_2, x_3) =
        > (0.5, 0.5, 1)$
        which is not a BFS.



      This confuses me. I was under the impression that extreme points are the same as basic solutions. Is this true?



      If it is true and if one of basic solutions is not the optimal solution, then what is even the point of calculating the basic solutions? What am I missing here? Any help to get out of this confusion is appreciated!







      linear-programming






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 1 '18 at 14:18







      PPGoodMan

















      asked Nov 1 '18 at 14:10









      PPGoodManPPGoodMan

      757




      757




















          1 Answer
          1






          active

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          1












          $begingroup$

          $(0.5,0.5,1)$ is not a BFS and it is also not an extreme point of $P$. There are only $2$ independent constraints that are active.



          Notice that $$min_x_1,x_2,x_3 max x_1 -0.5, -x_1+0.5$$



          subject to $$x_1+x_2=1$$
          $$x_3=1$$
          $$xge 0$$



          is not a linear programming problem.



          However, it can be converted to a linear programming problem.



          $$min_x_1,x_2,x_3,z z$$



          subject to
          $$z ge x_1 - 0.5$$
          $$z ge -x_1+0.5$$
          $$x_1+x_2=1$$
          $$x_3=1$$
          $$xge 0$$



          Notice that $4$ independent constraints are active. The BFS in the new feasible set is $(0.5,0.5,1,0)$






          share|cite|improve this answer









          $endgroup$













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            1 Answer
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            1 Answer
            1






            active

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            active

            oldest

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            active

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            1












            $begingroup$

            $(0.5,0.5,1)$ is not a BFS and it is also not an extreme point of $P$. There are only $2$ independent constraints that are active.



            Notice that $$min_x_1,x_2,x_3 max x_1 -0.5, -x_1+0.5$$



            subject to $$x_1+x_2=1$$
            $$x_3=1$$
            $$xge 0$$



            is not a linear programming problem.



            However, it can be converted to a linear programming problem.



            $$min_x_1,x_2,x_3,z z$$



            subject to
            $$z ge x_1 - 0.5$$
            $$z ge -x_1+0.5$$
            $$x_1+x_2=1$$
            $$x_3=1$$
            $$xge 0$$



            Notice that $4$ independent constraints are active. The BFS in the new feasible set is $(0.5,0.5,1,0)$






            share|cite|improve this answer









            $endgroup$

















              1












              $begingroup$

              $(0.5,0.5,1)$ is not a BFS and it is also not an extreme point of $P$. There are only $2$ independent constraints that are active.



              Notice that $$min_x_1,x_2,x_3 max x_1 -0.5, -x_1+0.5$$



              subject to $$x_1+x_2=1$$
              $$x_3=1$$
              $$xge 0$$



              is not a linear programming problem.



              However, it can be converted to a linear programming problem.



              $$min_x_1,x_2,x_3,z z$$



              subject to
              $$z ge x_1 - 0.5$$
              $$z ge -x_1+0.5$$
              $$x_1+x_2=1$$
              $$x_3=1$$
              $$xge 0$$



              Notice that $4$ independent constraints are active. The BFS in the new feasible set is $(0.5,0.5,1,0)$






              share|cite|improve this answer









              $endgroup$















                1












                1








                1





                $begingroup$

                $(0.5,0.5,1)$ is not a BFS and it is also not an extreme point of $P$. There are only $2$ independent constraints that are active.



                Notice that $$min_x_1,x_2,x_3 max x_1 -0.5, -x_1+0.5$$



                subject to $$x_1+x_2=1$$
                $$x_3=1$$
                $$xge 0$$



                is not a linear programming problem.



                However, it can be converted to a linear programming problem.



                $$min_x_1,x_2,x_3,z z$$



                subject to
                $$z ge x_1 - 0.5$$
                $$z ge -x_1+0.5$$
                $$x_1+x_2=1$$
                $$x_3=1$$
                $$xge 0$$



                Notice that $4$ independent constraints are active. The BFS in the new feasible set is $(0.5,0.5,1,0)$






                share|cite|improve this answer









                $endgroup$



                $(0.5,0.5,1)$ is not a BFS and it is also not an extreme point of $P$. There are only $2$ independent constraints that are active.



                Notice that $$min_x_1,x_2,x_3 max x_1 -0.5, -x_1+0.5$$



                subject to $$x_1+x_2=1$$
                $$x_3=1$$
                $$xge 0$$



                is not a linear programming problem.



                However, it can be converted to a linear programming problem.



                $$min_x_1,x_2,x_3,z z$$



                subject to
                $$z ge x_1 - 0.5$$
                $$z ge -x_1+0.5$$
                $$x_1+x_2=1$$
                $$x_3=1$$
                $$xge 0$$



                Notice that $4$ independent constraints are active. The BFS in the new feasible set is $(0.5,0.5,1,0)$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 1 '18 at 14:59









                Siong Thye GohSiong Thye Goh

                104k1468120




                104k1468120



























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