How to show that $min_{x in mathbb{R}^4} x^Tx$ subject to $x^TAx geq 1$ has a global minimizer?












1












$begingroup$


Consider the following problem:



$$min_{x in mathbb{R}^4} x^Tx$$



over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.



How to show that this problem has a global minimizer?



I need to show that there exist $x_*$ in $C$ for which I have
$$
x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
$$



or to come up with something like the following
$$
|x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
$$



My try:



$A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,



$$
x^TAx=x^Tu Lambda u^Tx geq 1
$$



So



$$
z^T Lambda z geq 1
$$
where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be



$$min_{x in mathbb{R}^4} z^Tz$$
over $z^T Lambda z geq 1$.
How can we proceed?






linear-algebra optimization symmetric-matrices







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Consider the following problem:



    $$min_{x in mathbb{R}^4} x^Tx$$



    over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.



    How to show that this problem has a global minimizer?



    I need to show that there exist $x_*$ in $C$ for which I have
    $$
    x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
    $$



    or to come up with something like the following
    $$
    |x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
    $$



    My try:



    $A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,



    $$
    x^TAx=x^Tu Lambda u^Tx geq 1
    $$



    So



    $$
    z^T Lambda z geq 1
    $$
    where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be



    $$min_{x in mathbb{R}^4} z^Tz$$
    over $z^T Lambda z geq 1$.
    How can we proceed?






    linear-algebra optimization symmetric-matrices







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Consider the following problem:



      $$min_{x in mathbb{R}^4} x^Tx$$



      over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.



      How to show that this problem has a global minimizer?



      I need to show that there exist $x_*$ in $C$ for which I have
      $$
      x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
      $$



      or to come up with something like the following
      $$
      |x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
      $$



      My try:



      $A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,



      $$
      x^TAx=x^Tu Lambda u^Tx geq 1
      $$



      So



      $$
      z^T Lambda z geq 1
      $$
      where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be



      $$min_{x in mathbb{R}^4} z^Tz$$
      over $z^T Lambda z geq 1$.
      How can we proceed?






      linear-algebra optimization symmetric-matrices







      share|cite|improve this question









      $endgroup$




      Consider the following problem:



      $$min_{x in mathbb{R}^4} x^Tx$$



      over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.



      How to show that this problem has a global minimizer?



      I need to show that there exist $x_*$ in $C$ for which I have
      $$
      x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
      $$



      or to come up with something like the following
      $$
      |x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
      $$



      My try:



      $A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,



      $$
      x^TAx=x^Tu Lambda u^Tx geq 1
      $$



      So



      $$
      z^T Lambda z geq 1
      $$
      where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be



      $$min_{x in mathbb{R}^4} z^Tz$$
      over $z^T Lambda z geq 1$.
      How can we proceed?






      linear-algebra optimization symmetric-matrices




      linear-algebra optimization symmetric-matrices






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 17 '18 at 1:40









      SepideSepide

      3038




      3038






















          1 Answer
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          1












          $begingroup$

          Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).



          Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            I do not know, can you explain it to me?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:08










          • $begingroup$
            @Sepide because any objective value outside B_R is surely >= inside B_R.
            $endgroup$
            – Vim
            Dec 17 '18 at 2:10










          • $begingroup$
            Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:22










          • $begingroup$
            @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
            $endgroup$
            – Vim
            Dec 17 '18 at 3:17











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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).



          Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            I do not know, can you explain it to me?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:08










          • $begingroup$
            @Sepide because any objective value outside B_R is surely >= inside B_R.
            $endgroup$
            – Vim
            Dec 17 '18 at 2:10










          • $begingroup$
            Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:22










          • $begingroup$
            @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
            $endgroup$
            – Vim
            Dec 17 '18 at 3:17
















          1












          $begingroup$

          Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).



          Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            I do not know, can you explain it to me?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:08










          • $begingroup$
            @Sepide because any objective value outside B_R is surely >= inside B_R.
            $endgroup$
            – Vim
            Dec 17 '18 at 2:10










          • $begingroup$
            Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:22










          • $begingroup$
            @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
            $endgroup$
            – Vim
            Dec 17 '18 at 3:17














          1












          1








          1





          $begingroup$

          Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).



          Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.






          share|cite|improve this answer











          $endgroup$



          Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).



          Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 17 '18 at 3:21

























          answered Dec 17 '18 at 1:47









          VimVim

          8,12031348




          8,12031348












          • $begingroup$
            I do not know, can you explain it to me?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:08










          • $begingroup$
            @Sepide because any objective value outside B_R is surely >= inside B_R.
            $endgroup$
            – Vim
            Dec 17 '18 at 2:10










          • $begingroup$
            Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:22










          • $begingroup$
            @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
            $endgroup$
            – Vim
            Dec 17 '18 at 3:17


















          • $begingroup$
            I do not know, can you explain it to me?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:08










          • $begingroup$
            @Sepide because any objective value outside B_R is surely >= inside B_R.
            $endgroup$
            – Vim
            Dec 17 '18 at 2:10










          • $begingroup$
            Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
            $endgroup$
            – Sepide
            Dec 17 '18 at 2:22










          • $begingroup$
            @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
            $endgroup$
            – Vim
            Dec 17 '18 at 3:17
















          $begingroup$
          I do not know, can you explain it to me?
          $endgroup$
          – Sepide
          Dec 17 '18 at 2:08




          $begingroup$
          I do not know, can you explain it to me?
          $endgroup$
          – Sepide
          Dec 17 '18 at 2:08












          $begingroup$
          @Sepide because any objective value outside B_R is surely >= inside B_R.
          $endgroup$
          – Vim
          Dec 17 '18 at 2:10




          $begingroup$
          @Sepide because any objective value outside B_R is surely >= inside B_R.
          $endgroup$
          – Vim
          Dec 17 '18 at 2:10












          $begingroup$
          Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
          $endgroup$
          – Sepide
          Dec 17 '18 at 2:22




          $begingroup$
          Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
          $endgroup$
          – Sepide
          Dec 17 '18 at 2:22












          $begingroup$
          @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
          $endgroup$
          – Vim
          Dec 17 '18 at 3:17




          $begingroup$
          @Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
          $endgroup$
          – Vim
          Dec 17 '18 at 3:17


















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