L2 constrained infinity norm minimization












1












$begingroup$


I have the following minimization problem:



$$||x||_{infty} rightarrow min$$
s.t. $$ ||x - x_0||^2_2 le r^2$$



where (to avoid trivial solutions) $||x_0||_2 > r $



Preliminary remarks:



-everything is convex i.e. we can use Lagrange dual



-if $x^{*}$ is a solution then $||x^{*} -x_0||_2 = r $



Unfortunately I have a great problem with estabilishing
$$inf_{x in mathbb{R}^n} ||x||_{infty} + mu cdot ||x- x_0||^2_2$$
(the $-mu r^2$ part does not affect calculations of infimum)
which is mandatory if I want to solve the dual problem.



If $||x||_{infty} = alpha<= ||x_0||_{infty}$ then
$$inf_{x : ||x||_{infty} = alpha} ||x||_{infty} + mu cdot ||x- x_0||^2_2 =
alpha + mu cdot || x_0 wedge alpha - x_0 ||_{2}^2$$



where $x_0 wedge alpha ^{(i)} := min(|x_0^{(i)}|,alpha) cdot sgn(x_0^{(i)})$



but further minimization with respect to alpha seems to be intractable.



Is there some way to obtain closed form expression of $L(mu)$ ?



I've also tried solving the primal problem by stating that
$x^{*}$ must be of the form (hope that's true):



$x^{*,(i)} = x_0^{(i)}$ for $i in I$
and
$x^{*,(j)} = t cdot x_0^{(j)}$ for $j in J$



where $t < 1 $ and $J cap I = emptyset$ and $I cup J = [1,2...,n]$



but I have problems characterizing sets $J,I$ in dimension biger than 2.



Any help greatly appreciated.






optimization convex-optimization norm nonlinear-optimization







share|cite|improve this question









$endgroup$












  • $begingroup$
    was my answer helpful?
    $endgroup$
    – LinAlg
    Feb 25 at 16:33
















1












$begingroup$


I have the following minimization problem:



$$||x||_{infty} rightarrow min$$
s.t. $$ ||x - x_0||^2_2 le r^2$$



where (to avoid trivial solutions) $||x_0||_2 > r $



Preliminary remarks:



-everything is convex i.e. we can use Lagrange dual



-if $x^{*}$ is a solution then $||x^{*} -x_0||_2 = r $



Unfortunately I have a great problem with estabilishing
$$inf_{x in mathbb{R}^n} ||x||_{infty} + mu cdot ||x- x_0||^2_2$$
(the $-mu r^2$ part does not affect calculations of infimum)
which is mandatory if I want to solve the dual problem.



If $||x||_{infty} = alpha<= ||x_0||_{infty}$ then
$$inf_{x : ||x||_{infty} = alpha} ||x||_{infty} + mu cdot ||x- x_0||^2_2 =
alpha + mu cdot || x_0 wedge alpha - x_0 ||_{2}^2$$



where $x_0 wedge alpha ^{(i)} := min(|x_0^{(i)}|,alpha) cdot sgn(x_0^{(i)})$



but further minimization with respect to alpha seems to be intractable.



Is there some way to obtain closed form expression of $L(mu)$ ?



I've also tried solving the primal problem by stating that
$x^{*}$ must be of the form (hope that's true):



$x^{*,(i)} = x_0^{(i)}$ for $i in I$
and
$x^{*,(j)} = t cdot x_0^{(j)}$ for $j in J$



where $t < 1 $ and $J cap I = emptyset$ and $I cup J = [1,2...,n]$



but I have problems characterizing sets $J,I$ in dimension biger than 2.



Any help greatly appreciated.






optimization convex-optimization norm nonlinear-optimization







share|cite|improve this question









$endgroup$












  • $begingroup$
    was my answer helpful?
    $endgroup$
    – LinAlg
    Feb 25 at 16:33














1












1








1


1



$begingroup$


I have the following minimization problem:



$$||x||_{infty} rightarrow min$$
s.t. $$ ||x - x_0||^2_2 le r^2$$



where (to avoid trivial solutions) $||x_0||_2 > r $



Preliminary remarks:



-everything is convex i.e. we can use Lagrange dual



-if $x^{*}$ is a solution then $||x^{*} -x_0||_2 = r $



Unfortunately I have a great problem with estabilishing
$$inf_{x in mathbb{R}^n} ||x||_{infty} + mu cdot ||x- x_0||^2_2$$
(the $-mu r^2$ part does not affect calculations of infimum)
which is mandatory if I want to solve the dual problem.



If $||x||_{infty} = alpha<= ||x_0||_{infty}$ then
$$inf_{x : ||x||_{infty} = alpha} ||x||_{infty} + mu cdot ||x- x_0||^2_2 =
alpha + mu cdot || x_0 wedge alpha - x_0 ||_{2}^2$$



where $x_0 wedge alpha ^{(i)} := min(|x_0^{(i)}|,alpha) cdot sgn(x_0^{(i)})$



but further minimization with respect to alpha seems to be intractable.



Is there some way to obtain closed form expression of $L(mu)$ ?



I've also tried solving the primal problem by stating that
$x^{*}$ must be of the form (hope that's true):



$x^{*,(i)} = x_0^{(i)}$ for $i in I$
and
$x^{*,(j)} = t cdot x_0^{(j)}$ for $j in J$



where $t < 1 $ and $J cap I = emptyset$ and $I cup J = [1,2...,n]$



but I have problems characterizing sets $J,I$ in dimension biger than 2.



Any help greatly appreciated.






optimization convex-optimization norm nonlinear-optimization







share|cite|improve this question









$endgroup$




I have the following minimization problem:



$$||x||_{infty} rightarrow min$$
s.t. $$ ||x - x_0||^2_2 le r^2$$



where (to avoid trivial solutions) $||x_0||_2 > r $



Preliminary remarks:



-everything is convex i.e. we can use Lagrange dual



-if $x^{*}$ is a solution then $||x^{*} -x_0||_2 = r $



Unfortunately I have a great problem with estabilishing
$$inf_{x in mathbb{R}^n} ||x||_{infty} + mu cdot ||x- x_0||^2_2$$
(the $-mu r^2$ part does not affect calculations of infimum)
which is mandatory if I want to solve the dual problem.



If $||x||_{infty} = alpha<= ||x_0||_{infty}$ then
$$inf_{x : ||x||_{infty} = alpha} ||x||_{infty} + mu cdot ||x- x_0||^2_2 =
alpha + mu cdot || x_0 wedge alpha - x_0 ||_{2}^2$$



where $x_0 wedge alpha ^{(i)} := min(|x_0^{(i)}|,alpha) cdot sgn(x_0^{(i)})$



but further minimization with respect to alpha seems to be intractable.



Is there some way to obtain closed form expression of $L(mu)$ ?



I've also tried solving the primal problem by stating that
$x^{*}$ must be of the form (hope that's true):



$x^{*,(i)} = x_0^{(i)}$ for $i in I$
and
$x^{*,(j)} = t cdot x_0^{(j)}$ for $j in J$



where $t < 1 $ and $J cap I = emptyset$ and $I cup J = [1,2...,n]$



but I have problems characterizing sets $J,I$ in dimension biger than 2.



Any help greatly appreciated.






optimization convex-optimization norm nonlinear-optimization




optimization convex-optimization norm nonlinear-optimization






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 5 at 12:26









marcusymarcusy

362




362












  • $begingroup$
    was my answer helpful?
    $endgroup$
    – LinAlg
    Feb 25 at 16:33


















  • $begingroup$
    was my answer helpful?
    $endgroup$
    – LinAlg
    Feb 25 at 16:33
















$begingroup$
was my answer helpful?
$endgroup$
– LinAlg
Feb 25 at 16:33




$begingroup$
was my answer helpful?
$endgroup$
– LinAlg
Feb 25 at 16:33










1 Answer
1






active

oldest

votes


















1












$begingroup$

Write the problem as
$$min_{x,y} {||x||_{infty} : ||y - x_0||^2_2 le r^2, x=y }$$



Then you only need to estabilish
$$min_{x,y in mathbb{R}^n} ||x||_{infty} + mu ||y- x_0||^2_2 +lambda^T(x-y)$$
This can be trivially related to the convex conjugates of the respective norms.






share|cite|improve this answer









$endgroup$













    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',
    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%2f3062664%2fl2-constrained-infinity-norm-minimization%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












    $begingroup$

    Write the problem as
    $$min_{x,y} {||x||_{infty} : ||y - x_0||^2_2 le r^2, x=y }$$



    Then you only need to estabilish
    $$min_{x,y in mathbb{R}^n} ||x||_{infty} + mu ||y- x_0||^2_2 +lambda^T(x-y)$$
    This can be trivially related to the convex conjugates of the respective norms.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Write the problem as
      $$min_{x,y} {||x||_{infty} : ||y - x_0||^2_2 le r^2, x=y }$$



      Then you only need to estabilish
      $$min_{x,y in mathbb{R}^n} ||x||_{infty} + mu ||y- x_0||^2_2 +lambda^T(x-y)$$
      This can be trivially related to the convex conjugates of the respective norms.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Write the problem as
        $$min_{x,y} {||x||_{infty} : ||y - x_0||^2_2 le r^2, x=y }$$



        Then you only need to estabilish
        $$min_{x,y in mathbb{R}^n} ||x||_{infty} + mu ||y- x_0||^2_2 +lambda^T(x-y)$$
        This can be trivially related to the convex conjugates of the respective norms.






        share|cite|improve this answer









        $endgroup$



        Write the problem as
        $$min_{x,y} {||x||_{infty} : ||y - x_0||^2_2 le r^2, x=y }$$



        Then you only need to estabilish
        $$min_{x,y in mathbb{R}^n} ||x||_{infty} + mu ||y- x_0||^2_2 +lambda^T(x-y)$$
        This can be trivially related to the convex conjugates of the respective norms.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 5 at 16:10









        LinAlgLinAlg

        10k1521




        10k1521






























            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%2f3062664%2fl2-constrained-infinity-norm-minimization%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

            Ellipse (mathématiques)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
            Read more