Proving a projection based on Farkas Lemma












0












$begingroup$


In an Integer Programming book, a proof is given for a theorem but i do not understand a certain step:



Theorem: Consider a polyhedron $P := {(x,z) in R^n times R^p: Ax+Bzleq b}$ and let $r^1$,...,$r^q$ be the extreme rays of $C_p$: the cone of $P$. Then, $proj_x(P) = {x in R^n: r^tAx leq r^t forall t = 1,...,q}$



Proof: It suffices to show that for any $x$, $ x notin proj_x(P) $ if and only if $r^tAx>r^tb$ for some $t = 1,...,q$. By definition $ x notin proj_x(P) $ if and only if the system $Bz <b-Ax$ is infeasible.



I get it until here



Proof (cont):
By Farkas Lemma, the latter system is infeasible if and only if there exists a vector $u in C_p$ such that $uAx geq ub$



Here is where i am lost. How do they come to this based on Farkas Lemma?



Farkas lemma:
Given matrix $A$ and vector $b$, exactly one of the following statements is true:





  1. $exists x$ such that $Ax=b$ and $x geq 0$


  2. $exists y$ such that $y^TA geq 0^T$ and $y^Tb < 0$


Definition Cone:
The cone of $P={(x,z) in R^n times R^p: Ax +Bz leq b}$ is $C_P = {u in R^m: uB=0, u geq 0}$






polyhedra projection







share|cite|improve this question











$endgroup$












  • $begingroup$
    Is this referring to the same question as math.stackexchange.com/questions/2968085/…? They look similar but the $R$'s are $R_+$'s and there is a $V$ instead of $C_p.$ What's the cone of $P$?
    $endgroup$
    – Dap
    Jan 21 at 6:55












  • $begingroup$
    @Dap It is based on theory I found while looking for an answer to that question. Not sure if it is 100% the same. Anyway, i added the definition of the cone.
    $endgroup$
    – user3053216
    Jan 21 at 9:21
















0












$begingroup$


In an Integer Programming book, a proof is given for a theorem but i do not understand a certain step:



Theorem: Consider a polyhedron $P := {(x,z) in R^n times R^p: Ax+Bzleq b}$ and let $r^1$,...,$r^q$ be the extreme rays of $C_p$: the cone of $P$. Then, $proj_x(P) = {x in R^n: r^tAx leq r^t forall t = 1,...,q}$



Proof: It suffices to show that for any $x$, $ x notin proj_x(P) $ if and only if $r^tAx>r^tb$ for some $t = 1,...,q$. By definition $ x notin proj_x(P) $ if and only if the system $Bz <b-Ax$ is infeasible.



I get it until here



Proof (cont):
By Farkas Lemma, the latter system is infeasible if and only if there exists a vector $u in C_p$ such that $uAx geq ub$



Here is where i am lost. How do they come to this based on Farkas Lemma?



Farkas lemma:
Given matrix $A$ and vector $b$, exactly one of the following statements is true:





  1. $exists x$ such that $Ax=b$ and $x geq 0$


  2. $exists y$ such that $y^TA geq 0^T$ and $y^Tb < 0$


Definition Cone:
The cone of $P={(x,z) in R^n times R^p: Ax +Bz leq b}$ is $C_P = {u in R^m: uB=0, u geq 0}$






polyhedra projection







share|cite|improve this question











$endgroup$












  • $begingroup$
    Is this referring to the same question as math.stackexchange.com/questions/2968085/…? They look similar but the $R$'s are $R_+$'s and there is a $V$ instead of $C_p.$ What's the cone of $P$?
    $endgroup$
    – Dap
    Jan 21 at 6:55












  • $begingroup$
    @Dap It is based on theory I found while looking for an answer to that question. Not sure if it is 100% the same. Anyway, i added the definition of the cone.
    $endgroup$
    – user3053216
    Jan 21 at 9:21














0












0








0





$begingroup$


In an Integer Programming book, a proof is given for a theorem but i do not understand a certain step:



Theorem: Consider a polyhedron $P := {(x,z) in R^n times R^p: Ax+Bzleq b}$ and let $r^1$,...,$r^q$ be the extreme rays of $C_p$: the cone of $P$. Then, $proj_x(P) = {x in R^n: r^tAx leq r^t forall t = 1,...,q}$



Proof: It suffices to show that for any $x$, $ x notin proj_x(P) $ if and only if $r^tAx>r^tb$ for some $t = 1,...,q$. By definition $ x notin proj_x(P) $ if and only if the system $Bz <b-Ax$ is infeasible.



I get it until here



Proof (cont):
By Farkas Lemma, the latter system is infeasible if and only if there exists a vector $u in C_p$ such that $uAx geq ub$



Here is where i am lost. How do they come to this based on Farkas Lemma?



Farkas lemma:
Given matrix $A$ and vector $b$, exactly one of the following statements is true:





  1. $exists x$ such that $Ax=b$ and $x geq 0$


  2. $exists y$ such that $y^TA geq 0^T$ and $y^Tb < 0$


Definition Cone:
The cone of $P={(x,z) in R^n times R^p: Ax +Bz leq b}$ is $C_P = {u in R^m: uB=0, u geq 0}$






polyhedra projection







share|cite|improve this question











$endgroup$




In an Integer Programming book, a proof is given for a theorem but i do not understand a certain step:



Theorem: Consider a polyhedron $P := {(x,z) in R^n times R^p: Ax+Bzleq b}$ and let $r^1$,...,$r^q$ be the extreme rays of $C_p$: the cone of $P$. Then, $proj_x(P) = {x in R^n: r^tAx leq r^t forall t = 1,...,q}$



Proof: It suffices to show that for any $x$, $ x notin proj_x(P) $ if and only if $r^tAx>r^tb$ for some $t = 1,...,q$. By definition $ x notin proj_x(P) $ if and only if the system $Bz <b-Ax$ is infeasible.



I get it until here



Proof (cont):
By Farkas Lemma, the latter system is infeasible if and only if there exists a vector $u in C_p$ such that $uAx geq ub$



Here is where i am lost. How do they come to this based on Farkas Lemma?



Farkas lemma:
Given matrix $A$ and vector $b$, exactly one of the following statements is true:





  1. $exists x$ such that $Ax=b$ and $x geq 0$


  2. $exists y$ such that $y^TA geq 0^T$ and $y^Tb < 0$


Definition Cone:
The cone of $P={(x,z) in R^n times R^p: Ax +Bz leq b}$ is $C_P = {u in R^m: uB=0, u geq 0}$






polyhedra projection




polyhedra projection






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 21 at 10:34







user3053216

















asked Jan 18 at 21:49









user3053216user3053216

13111




13111












  • $begingroup$
    Is this referring to the same question as math.stackexchange.com/questions/2968085/…? They look similar but the $R$'s are $R_+$'s and there is a $V$ instead of $C_p.$ What's the cone of $P$?
    $endgroup$
    – Dap
    Jan 21 at 6:55












  • $begingroup$
    @Dap It is based on theory I found while looking for an answer to that question. Not sure if it is 100% the same. Anyway, i added the definition of the cone.
    $endgroup$
    – user3053216
    Jan 21 at 9:21


















  • $begingroup$
    Is this referring to the same question as math.stackexchange.com/questions/2968085/…? They look similar but the $R$'s are $R_+$'s and there is a $V$ instead of $C_p.$ What's the cone of $P$?
    $endgroup$
    – Dap
    Jan 21 at 6:55












  • $begingroup$
    @Dap It is based on theory I found while looking for an answer to that question. Not sure if it is 100% the same. Anyway, i added the definition of the cone.
    $endgroup$
    – user3053216
    Jan 21 at 9:21
















$begingroup$
Is this referring to the same question as math.stackexchange.com/questions/2968085/…? They look similar but the $R$'s are $R_+$'s and there is a $V$ instead of $C_p.$ What's the cone of $P$?
$endgroup$
– Dap
Jan 21 at 6:55






$begingroup$
Is this referring to the same question as math.stackexchange.com/questions/2968085/…? They look similar but the $R$'s are $R_+$'s and there is a $V$ instead of $C_p.$ What's the cone of $P$?
$endgroup$
– Dap
Jan 21 at 6:55














$begingroup$
@Dap It is based on theory I found while looking for an answer to that question. Not sure if it is 100% the same. Anyway, i added the definition of the cone.
$endgroup$
– user3053216
Jan 21 at 9:21




$begingroup$
@Dap It is based on theory I found while looking for an answer to that question. Not sure if it is 100% the same. Anyway, i added the definition of the cone.
$endgroup$
– user3053216
Jan 21 at 9:21










1 Answer
1






active

oldest

votes


















1





+50







$begingroup$

$xnotinmathrm{proj}_x(P)$ if and only if the system $Bzleq b-Ax$ is infeasible.



The statement you want follows from one of the generalized forms of Farkas' lemma. In particular Theorem 3.6 in "Integer Programming" by Conforti et al says that exactly one of these holds:





  • $exists y$ such that $Byleq f$


  • $exists u$ such that $uf< 0$ and $uB=0,$ $ugeq 0$


Substituting in $z$ and $b-Ax$ for $y$ and $f$ gives what you want.




To prove the generalization from the basic version of Farkas' lemma, add slack variables $w$ to convert the system to an equality, and split $y$ into positive and negative parts:



$$exists y^+,y^-,wtext{ such that }By-By+w=ftext{ and }y^+,y^-,wgeq 0$$
or in block matrix form:
$$exists y^+,y^-,wtext{ such that }begin{pmatrix}B&-B&Iend{pmatrix}begin{pmatrix}y^+\y^-\wend{pmatrix}=ftext{ and }y,y',wgeq 0$$



By Farkas' lemma this system is infeasible if and only if:
$$exists utext{ such that }ubegin{pmatrix}B&-B&Iend{pmatrix}geq 0text{ and }uf<0$$
i.e.
$$exists utext{ such that }uB=0text{ and }ugeq 0text{ and }uf<0.$$






share|cite|improve this answer









$endgroup$














    Your Answer








    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',
    autoActivateHeartbeat: false,
    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%2f3078797%2fproving-a-projection-based-on-farkas-lemma%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









    1





    +50







    $begingroup$

    $xnotinmathrm{proj}_x(P)$ if and only if the system $Bzleq b-Ax$ is infeasible.



    The statement you want follows from one of the generalized forms of Farkas' lemma. In particular Theorem 3.6 in "Integer Programming" by Conforti et al says that exactly one of these holds:





    • $exists y$ such that $Byleq f$


    • $exists u$ such that $uf< 0$ and $uB=0,$ $ugeq 0$


    Substituting in $z$ and $b-Ax$ for $y$ and $f$ gives what you want.




    To prove the generalization from the basic version of Farkas' lemma, add slack variables $w$ to convert the system to an equality, and split $y$ into positive and negative parts:



    $$exists y^+,y^-,wtext{ such that }By-By+w=ftext{ and }y^+,y^-,wgeq 0$$
    or in block matrix form:
    $$exists y^+,y^-,wtext{ such that }begin{pmatrix}B&-B&Iend{pmatrix}begin{pmatrix}y^+\y^-\wend{pmatrix}=ftext{ and }y,y',wgeq 0$$



    By Farkas' lemma this system is infeasible if and only if:
    $$exists utext{ such that }ubegin{pmatrix}B&-B&Iend{pmatrix}geq 0text{ and }uf<0$$
    i.e.
    $$exists utext{ such that }uB=0text{ and }ugeq 0text{ and }uf<0.$$






    share|cite|improve this answer









    $endgroup$


















      1





      +50







      $begingroup$

      $xnotinmathrm{proj}_x(P)$ if and only if the system $Bzleq b-Ax$ is infeasible.



      The statement you want follows from one of the generalized forms of Farkas' lemma. In particular Theorem 3.6 in "Integer Programming" by Conforti et al says that exactly one of these holds:





      • $exists y$ such that $Byleq f$


      • $exists u$ such that $uf< 0$ and $uB=0,$ $ugeq 0$


      Substituting in $z$ and $b-Ax$ for $y$ and $f$ gives what you want.




      To prove the generalization from the basic version of Farkas' lemma, add slack variables $w$ to convert the system to an equality, and split $y$ into positive and negative parts:



      $$exists y^+,y^-,wtext{ such that }By-By+w=ftext{ and }y^+,y^-,wgeq 0$$
      or in block matrix form:
      $$exists y^+,y^-,wtext{ such that }begin{pmatrix}B&-B&Iend{pmatrix}begin{pmatrix}y^+\y^-\wend{pmatrix}=ftext{ and }y,y',wgeq 0$$



      By Farkas' lemma this system is infeasible if and only if:
      $$exists utext{ such that }ubegin{pmatrix}B&-B&Iend{pmatrix}geq 0text{ and }uf<0$$
      i.e.
      $$exists utext{ such that }uB=0text{ and }ugeq 0text{ and }uf<0.$$






      share|cite|improve this answer









      $endgroup$
















        1





        +50







        1





        +50



        1




        +50



        $begingroup$

        $xnotinmathrm{proj}_x(P)$ if and only if the system $Bzleq b-Ax$ is infeasible.



        The statement you want follows from one of the generalized forms of Farkas' lemma. In particular Theorem 3.6 in "Integer Programming" by Conforti et al says that exactly one of these holds:





        • $exists y$ such that $Byleq f$


        • $exists u$ such that $uf< 0$ and $uB=0,$ $ugeq 0$


        Substituting in $z$ and $b-Ax$ for $y$ and $f$ gives what you want.




        To prove the generalization from the basic version of Farkas' lemma, add slack variables $w$ to convert the system to an equality, and split $y$ into positive and negative parts:



        $$exists y^+,y^-,wtext{ such that }By-By+w=ftext{ and }y^+,y^-,wgeq 0$$
        or in block matrix form:
        $$exists y^+,y^-,wtext{ such that }begin{pmatrix}B&-B&Iend{pmatrix}begin{pmatrix}y^+\y^-\wend{pmatrix}=ftext{ and }y,y',wgeq 0$$



        By Farkas' lemma this system is infeasible if and only if:
        $$exists utext{ such that }ubegin{pmatrix}B&-B&Iend{pmatrix}geq 0text{ and }uf<0$$
        i.e.
        $$exists utext{ such that }uB=0text{ and }ugeq 0text{ and }uf<0.$$






        share|cite|improve this answer









        $endgroup$



        $xnotinmathrm{proj}_x(P)$ if and only if the system $Bzleq b-Ax$ is infeasible.



        The statement you want follows from one of the generalized forms of Farkas' lemma. In particular Theorem 3.6 in "Integer Programming" by Conforti et al says that exactly one of these holds:





        • $exists y$ such that $Byleq f$


        • $exists u$ such that $uf< 0$ and $uB=0,$ $ugeq 0$


        Substituting in $z$ and $b-Ax$ for $y$ and $f$ gives what you want.




        To prove the generalization from the basic version of Farkas' lemma, add slack variables $w$ to convert the system to an equality, and split $y$ into positive and negative parts:



        $$exists y^+,y^-,wtext{ such that }By-By+w=ftext{ and }y^+,y^-,wgeq 0$$
        or in block matrix form:
        $$exists y^+,y^-,wtext{ such that }begin{pmatrix}B&-B&Iend{pmatrix}begin{pmatrix}y^+\y^-\wend{pmatrix}=ftext{ and }y,y',wgeq 0$$



        By Farkas' lemma this system is infeasible if and only if:
        $$exists utext{ such that }ubegin{pmatrix}B&-B&Iend{pmatrix}geq 0text{ and }uf<0$$
        i.e.
        $$exists utext{ such that }uB=0text{ and }ugeq 0text{ and }uf<0.$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 23 at 5:56









        DapDap

        20.2k842




        20.2k842






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078797%2fproving-a-projection-based-on-farkas-lemma%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

            Human spaceflight

            Image
            Human spaceflight From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Space traveler" redirects here. For other uses, see Space traveler (disambiguation). @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}} Apollo 11 crewmember Buzz Aldrin on the Moon, 1969 International Space Station crewmember Tracy Caldwell Dyson views the Earth, 2010 Space Shuttle Discovery heads into space with a crew aboard, STS-121 in 2006 Inside a space suit on the Canadarm, 1993 Human spaceflight (also referred to as crewed spaceflight or manned spaceflight ) is space travel with a crew or passengers aboard the spacecraft. Spacecraft carrying people may be operated directly, by human crew, or it may be e...
            Read more

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            Image
            0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
            Read more

            張江高科駅

            Image
            張江高科駅 ちょうこうこうか Zhangjiang Hi-Tech Park 所在地 中国上海市浦東新区 所属事業者 上海地鉄運営有限公司（上海軌道交通・駅詳細） 上海浦東現代有軌交通有限公司（駅詳細） テンプレートを表示 張江高科駅 各種表記 繁体字： 張江高科站 簡体字： 张江高科站 拼音 ： Zhàngjiāng Gāokē Zhàn 発音： ヂャンジャン ガオクー ヂャン 英文： Zhangjiang Hi-Tech Park Station テンプレートを表示 張江高科駅 （ちょうこうこうか-えき）は中華人民共和国上海市浦東新区に位置する、上海軌道交通2号線の駅である。 なお、張江高科駅前にある張江軌道電車の 張江地鉄駅駅 についてもここで述べる。 目次 1 利用可能な鉄道路線 2 沿革 3 駅構造 3.1 上海軌道交通 3.2 張江軌道電車 4 駅周辺 5 隣の駅 6 注釈 7 関連項目 利用可能な鉄道路線 上海軌道交通 ■ 2号線 上海浦東現代有軌交通有限公司 ■ 張江軌道電車 沿革 高架駅時代の姿 2000年12月27日 - 2号線の駅が高架駅として開業。 2010年 1月1日 - 張江軌道電車の開業に伴い、張江地鉄駅駅（停留所）を設置。 2月13日 - 駅地下化に伴い高架駅の運用停止。 2月24日 - 10日間の閉鎖を経て地下新駅の開業。 2017年2月9日 - リニューアル工事を経て再開業。乗降客の増加に対応し、ホームへのエレベーター、エスカレーターを増設、券売機エリアを拡張した [1] 。 駅構造 上海軌道交通 上海軌道交通 張江高科駅 5号口 ちょうこうこうか Zhangjiang Hi-Tech Park ◄ 竜陽路 (2.60km) (1.61km) 金科路 ► 所在地 中国上海市浦東新区松涛路 祖冲之路 （中国語版） 北緯31度12分11秒 東経121度34分50秒  /  北緯31.2...
            Read more