Solve $argmin_{x in mathbb{R}^n} I_C(x) + alpha sum_i left|y_i -x right|_2^2$, where $I_C(x)$: indicator...












0












$begingroup$


I have a problem on hand that I am not sure how to handle it. Can you please help/guide me to solve this optimization problem?



begin{align}
argmin_{x in mathbb{R}^n} I_C(x) + alpha sum_i left|y_i -x right|_2^2 ,
end{align}
where $I_C(x)$ is an indicator or characteristic function and the set $C$ can be a norm-2 ball, i.e., $C = left{x in mathbb{R}^n : left|x - c right|_2^2 leq rright}$, $y_i, c in mathbb{R}^n$ are given, and $r in mathbb{R}$.




My partial attempt:



The above problem can be rewritten as
begin{equation} label{eqn:weighted_projection_problem_definition_3}
begin{aligned}
& underset{x}{text{minimize}}
& & alpha sum_i left|y_i -x right|_2^2 \
& text{subject to}
& & left|x - c right|_2^2 leq r ,\
%&&& X succeq 0.
end{aligned}
end{equation}



Right?



If yes, then I think it is possible to solve in the closed form.






optimization convex-optimization







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    I have a problem on hand that I am not sure how to handle it. Can you please help/guide me to solve this optimization problem?



    begin{align}
    argmin_{x in mathbb{R}^n} I_C(x) + alpha sum_i left|y_i -x right|_2^2 ,
    end{align}
    where $I_C(x)$ is an indicator or characteristic function and the set $C$ can be a norm-2 ball, i.e., $C = left{x in mathbb{R}^n : left|x - c right|_2^2 leq rright}$, $y_i, c in mathbb{R}^n$ are given, and $r in mathbb{R}$.




    My partial attempt:



    The above problem can be rewritten as
    begin{equation} label{eqn:weighted_projection_problem_definition_3}
    begin{aligned}
    & underset{x}{text{minimize}}
    & & alpha sum_i left|y_i -x right|_2^2 \
    & text{subject to}
    & & left|x - c right|_2^2 leq r ,\
    %&&& X succeq 0.
    end{aligned}
    end{equation}



    Right?



    If yes, then I think it is possible to solve in the closed form.






    optimization convex-optimization







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I have a problem on hand that I am not sure how to handle it. Can you please help/guide me to solve this optimization problem?



      begin{align}
      argmin_{x in mathbb{R}^n} I_C(x) + alpha sum_i left|y_i -x right|_2^2 ,
      end{align}
      where $I_C(x)$ is an indicator or characteristic function and the set $C$ can be a norm-2 ball, i.e., $C = left{x in mathbb{R}^n : left|x - c right|_2^2 leq rright}$, $y_i, c in mathbb{R}^n$ are given, and $r in mathbb{R}$.




      My partial attempt:



      The above problem can be rewritten as
      begin{equation} label{eqn:weighted_projection_problem_definition_3}
      begin{aligned}
      & underset{x}{text{minimize}}
      & & alpha sum_i left|y_i -x right|_2^2 \
      & text{subject to}
      & & left|x - c right|_2^2 leq r ,\
      %&&& X succeq 0.
      end{aligned}
      end{equation}



      Right?



      If yes, then I think it is possible to solve in the closed form.






      optimization convex-optimization







      share|cite|improve this question











      $endgroup$




      I have a problem on hand that I am not sure how to handle it. Can you please help/guide me to solve this optimization problem?



      begin{align}
      argmin_{x in mathbb{R}^n} I_C(x) + alpha sum_i left|y_i -x right|_2^2 ,
      end{align}
      where $I_C(x)$ is an indicator or characteristic function and the set $C$ can be a norm-2 ball, i.e., $C = left{x in mathbb{R}^n : left|x - c right|_2^2 leq rright}$, $y_i, c in mathbb{R}^n$ are given, and $r in mathbb{R}$.




      My partial attempt:



      The above problem can be rewritten as
      begin{equation} label{eqn:weighted_projection_problem_definition_3}
      begin{aligned}
      & underset{x}{text{minimize}}
      & & alpha sum_i left|y_i -x right|_2^2 \
      & text{subject to}
      & & left|x - c right|_2^2 leq r ,\
      %&&& X succeq 0.
      end{aligned}
      end{equation}



      Right?



      If yes, then I think it is possible to solve in the closed form.






      optimization convex-optimization




      optimization convex-optimization






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 31 '18 at 11:43







      user550103

















      asked Dec 31 '18 at 9:20









      user550103user550103

      7581315




      7581315






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Let's complete the square: $| y_i - x |^2 = |y_i|^2 - 2 langle y_i, x rangle + |x|^2$, so
          begin{align}
          sum_{i=1}^N |y_i - x |^2 &= -2 left langle sum_{i=1}^N y_i, x rightrangle + N |x|^2 + ldots \
          &=N left(-2 left langle frac{1}{N}sum_{i=1}^N y_i, x rightrangle + |x|^2 right) + ldots \
          &= N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 + ldots
          end{align}
          where the ellipses ($ldots$) indicate terms that do not depend on $x$. It follows that
          begin{align}
          argmin_x , I_C(x) + alpha sum_{i=1}^N |y_i - x |^2&=
          argmin_x I_C(x) + alpha N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 \
          &= text{projection of } frac{1}{N} sum_{i=1}^N y_i text{ onto } C.
          end{align}






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you so much. I was thinking that it should boil down to projection problem...
            $endgroup$
            – user550103
            Dec 31 '18 at 13:16











          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%2f3057538%2fsolve-arg-min-x-in-mathbbrn-i-cx-alpha-sum-i-left-y-i-x-right%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












          $begingroup$

          Let's complete the square: $| y_i - x |^2 = |y_i|^2 - 2 langle y_i, x rangle + |x|^2$, so
          begin{align}
          sum_{i=1}^N |y_i - x |^2 &= -2 left langle sum_{i=1}^N y_i, x rightrangle + N |x|^2 + ldots \
          &=N left(-2 left langle frac{1}{N}sum_{i=1}^N y_i, x rightrangle + |x|^2 right) + ldots \
          &= N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 + ldots
          end{align}
          where the ellipses ($ldots$) indicate terms that do not depend on $x$. It follows that
          begin{align}
          argmin_x , I_C(x) + alpha sum_{i=1}^N |y_i - x |^2&=
          argmin_x I_C(x) + alpha N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 \
          &= text{projection of } frac{1}{N} sum_{i=1}^N y_i text{ onto } C.
          end{align}






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you so much. I was thinking that it should boil down to projection problem...
            $endgroup$
            – user550103
            Dec 31 '18 at 13:16
















          1












          $begingroup$

          Let's complete the square: $| y_i - x |^2 = |y_i|^2 - 2 langle y_i, x rangle + |x|^2$, so
          begin{align}
          sum_{i=1}^N |y_i - x |^2 &= -2 left langle sum_{i=1}^N y_i, x rightrangle + N |x|^2 + ldots \
          &=N left(-2 left langle frac{1}{N}sum_{i=1}^N y_i, x rightrangle + |x|^2 right) + ldots \
          &= N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 + ldots
          end{align}
          where the ellipses ($ldots$) indicate terms that do not depend on $x$. It follows that
          begin{align}
          argmin_x , I_C(x) + alpha sum_{i=1}^N |y_i - x |^2&=
          argmin_x I_C(x) + alpha N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 \
          &= text{projection of } frac{1}{N} sum_{i=1}^N y_i text{ onto } C.
          end{align}






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you so much. I was thinking that it should boil down to projection problem...
            $endgroup$
            – user550103
            Dec 31 '18 at 13:16














          1












          1








          1





          $begingroup$

          Let's complete the square: $| y_i - x |^2 = |y_i|^2 - 2 langle y_i, x rangle + |x|^2$, so
          begin{align}
          sum_{i=1}^N |y_i - x |^2 &= -2 left langle sum_{i=1}^N y_i, x rightrangle + N |x|^2 + ldots \
          &=N left(-2 left langle frac{1}{N}sum_{i=1}^N y_i, x rightrangle + |x|^2 right) + ldots \
          &= N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 + ldots
          end{align}
          where the ellipses ($ldots$) indicate terms that do not depend on $x$. It follows that
          begin{align}
          argmin_x , I_C(x) + alpha sum_{i=1}^N |y_i - x |^2&=
          argmin_x I_C(x) + alpha N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 \
          &= text{projection of } frac{1}{N} sum_{i=1}^N y_i text{ onto } C.
          end{align}






          share|cite|improve this answer











          $endgroup$



          Let's complete the square: $| y_i - x |^2 = |y_i|^2 - 2 langle y_i, x rangle + |x|^2$, so
          begin{align}
          sum_{i=1}^N |y_i - x |^2 &= -2 left langle sum_{i=1}^N y_i, x rightrangle + N |x|^2 + ldots \
          &=N left(-2 left langle frac{1}{N}sum_{i=1}^N y_i, x rightrangle + |x|^2 right) + ldots \
          &= N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 + ldots
          end{align}
          where the ellipses ($ldots$) indicate terms that do not depend on $x$. It follows that
          begin{align}
          argmin_x , I_C(x) + alpha sum_{i=1}^N |y_i - x |^2&=
          argmin_x I_C(x) + alpha N left|x - frac{1}{N} sum_{i=1}^N y_i right|^2 \
          &= text{projection of } frac{1}{N} sum_{i=1}^N y_i text{ onto } C.
          end{align}







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 31 '18 at 12:06

























          answered Dec 31 '18 at 12:00









          littleOlittleO

          29.9k646110




          29.9k646110












          • $begingroup$
            Thank you so much. I was thinking that it should boil down to projection problem...
            $endgroup$
            – user550103
            Dec 31 '18 at 13:16


















          • $begingroup$
            Thank you so much. I was thinking that it should boil down to projection problem...
            $endgroup$
            – user550103
            Dec 31 '18 at 13:16
















          $begingroup$
          Thank you so much. I was thinking that it should boil down to projection problem...
          $endgroup$
          – user550103
          Dec 31 '18 at 13:16




          $begingroup$
          Thank you so much. I was thinking that it should boil down to projection problem...
          $endgroup$
          – user550103
          Dec 31 '18 at 13:16


















          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%2f3057538%2fsolve-arg-min-x-in-mathbbrn-i-cx-alpha-sum-i-left-y-i-x-right%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

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more