Finding the Dual of an LP, where variables are not well isolated.











up vote
0
down vote

favorite












Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.



$$text{max } z
\
text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
\
x_1 + ldots + x_m = 1
\
x ge vec{0}
$$



, where $A_{.j}$ represents the j'th column of A.



My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
So I can't see how I can apply the standard primal-dual conversion.



Thanks!






optimization linear-programming duality-theorems







share|cite|improve this question


























    up vote
    0
    down vote

    favorite












    Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.



    $$text{max } z
    \
    text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
    \
    x_1 + ldots + x_m = 1
    \
    x ge vec{0}
    $$



    , where $A_{.j}$ represents the j'th column of A.



    My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
    So I can't see how I can apply the standard primal-dual conversion.



    Thanks!






    optimization linear-programming duality-theorems







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.



      $$text{max } z
      \
      text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
      \
      x_1 + ldots + x_m = 1
      \
      x ge vec{0}
      $$



      , where $A_{.j}$ represents the j'th column of A.



      My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
      So I can't see how I can apply the standard primal-dual conversion.



      Thanks!






      optimization linear-programming duality-theorems







      share|cite|improve this question













      Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.



      $$text{max } z
      \
      text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
      \
      x_1 + ldots + x_m = 1
      \
      x ge vec{0}
      $$



      , where $A_{.j}$ represents the j'th column of A.



      My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
      So I can't see how I can apply the standard primal-dual conversion.



      Thanks!






      optimization linear-programming duality-theorems




      optimization linear-programming duality-theorems






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 days ago









      Bojack Horseman

      1116




      1116






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
          $$begin{align}
          max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
          text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
          & x' geq 0.
          end{align}$$
          Deriving the dual is now simple.






          share|cite|improve this answer





















            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














             

            draft saved


            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005642%2ffinding-the-dual-of-an-lp-where-variables-are-not-well-isolated%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote



            accepted










            You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
            $$begin{align}
            max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
            text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
            & x' geq 0.
            end{align}$$
            Deriving the dual is now simple.






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
              $$begin{align}
              max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
              text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
              & x' geq 0.
              end{align}$$
              Deriving the dual is now simple.






              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
                $$begin{align}
                max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
                text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
                & x' geq 0.
                end{align}$$
                Deriving the dual is now simple.






                share|cite|improve this answer












                You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
                $$begin{align}
                max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
                text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
                & x' geq 0.
                end{align}$$
                Deriving the dual is now simple.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 days ago









                LinAlg

                7,6141520




                7,6141520






























                     

                    draft saved


                    draft discarded



















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005642%2ffinding-the-dual-of-an-lp-where-variables-are-not-well-isolated%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

                    Image
                    28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
                    Read more

                    Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

                    Image
                    up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
                    Read more

                    A Topological Invariant for $pi_3(U(n))$

                    Image
                    3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
                    Read more