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
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          $begingroup$

          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











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          $begingroup$

          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


















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