Large and ill-conditioned quadratic convex problem











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I need to solve a convex quadratic problem numerically:



$min f(x) = frac{1}{2} x^top A x - b^top x$,



where $A$ is a very large and ill-conditioned semi positive definite matrix. Typical conjugate gradient method doesn't work well. SGD is too slow.



I'm looking for a good numerical method with relatively less computational effort.



Any experience or suggestions? Thank you!










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    up vote
    1
    down vote

    favorite












    I need to solve a convex quadratic problem numerically:



    $min f(x) = frac{1}{2} x^top A x - b^top x$,



    where $A$ is a very large and ill-conditioned semi positive definite matrix. Typical conjugate gradient method doesn't work well. SGD is too slow.



    I'm looking for a good numerical method with relatively less computational effort.



    Any experience or suggestions? Thank you!










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I need to solve a convex quadratic problem numerically:



      $min f(x) = frac{1}{2} x^top A x - b^top x$,



      where $A$ is a very large and ill-conditioned semi positive definite matrix. Typical conjugate gradient method doesn't work well. SGD is too slow.



      I'm looking for a good numerical method with relatively less computational effort.



      Any experience or suggestions? Thank you!










      share|cite|improve this question















      I need to solve a convex quadratic problem numerically:



      $min f(x) = frac{1}{2} x^top A x - b^top x$,



      where $A$ is a very large and ill-conditioned semi positive definite matrix. Typical conjugate gradient method doesn't work well. SGD is too slow.



      I'm looking for a good numerical method with relatively less computational effort.



      Any experience or suggestions? Thank you!







      numerical-linear-algebra






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      share|cite|improve this question













      share|cite|improve this question




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      edited Nov 18 at 1:23

























      asked Nov 17 at 21:11









      Joe Lacey

      112




      112






















          1 Answer
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          I assume you are looking to minimize $f(x)$, and that $A$ is symmetric and positive semi-definite.




          • If $b notin Null(A)^{perp}$ then there is a vector $v in Null(A)$ such that $b^Tv neq 0$. Without loss of generality assume $b^Tv >0$ (else multiply $v$ by $-1$). Then for all real numbers $theta$ we get:
            $$ f(theta v) = 0 + (theta)(-b^Tv)$$
            so $lim_{thetarightarrowinfty} f(theta v) = -infty$ and no minimizer exists.



          • Else, $b in Null(A)^{perp} = Col(A^T) = Col(A)$. Just find any vector $r$ such that $Ar=b$. Then:
            begin{align}
            frac{1}{2}(x-r)^TA(x-r) &= frac{1}{2}x^TAx - x^TAr + frac{1}{2}r^TAr \
            &= f(x) + frac{1}{2}r^TAr
            end{align}

            and this is clearly minimized at $x^*=r$, so
            $$ f(x^*) = f(r) = -frac{1}{2}r^TAr$$







          In other words, the problem reduces to Gaussian elimination: Find a vector $r$ such that $Ar=b$. If no such $r$ exists then no minimizer exists.






          share|cite|improve this answer























          • I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
            – Joe Lacey
            Nov 18 at 1:25











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          up vote
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          down vote













          I assume you are looking to minimize $f(x)$, and that $A$ is symmetric and positive semi-definite.




          • If $b notin Null(A)^{perp}$ then there is a vector $v in Null(A)$ such that $b^Tv neq 0$. Without loss of generality assume $b^Tv >0$ (else multiply $v$ by $-1$). Then for all real numbers $theta$ we get:
            $$ f(theta v) = 0 + (theta)(-b^Tv)$$
            so $lim_{thetarightarrowinfty} f(theta v) = -infty$ and no minimizer exists.



          • Else, $b in Null(A)^{perp} = Col(A^T) = Col(A)$. Just find any vector $r$ such that $Ar=b$. Then:
            begin{align}
            frac{1}{2}(x-r)^TA(x-r) &= frac{1}{2}x^TAx - x^TAr + frac{1}{2}r^TAr \
            &= f(x) + frac{1}{2}r^TAr
            end{align}

            and this is clearly minimized at $x^*=r$, so
            $$ f(x^*) = f(r) = -frac{1}{2}r^TAr$$







          In other words, the problem reduces to Gaussian elimination: Find a vector $r$ such that $Ar=b$. If no such $r$ exists then no minimizer exists.






          share|cite|improve this answer























          • I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
            – Joe Lacey
            Nov 18 at 1:25















          up vote
          0
          down vote













          I assume you are looking to minimize $f(x)$, and that $A$ is symmetric and positive semi-definite.




          • If $b notin Null(A)^{perp}$ then there is a vector $v in Null(A)$ such that $b^Tv neq 0$. Without loss of generality assume $b^Tv >0$ (else multiply $v$ by $-1$). Then for all real numbers $theta$ we get:
            $$ f(theta v) = 0 + (theta)(-b^Tv)$$
            so $lim_{thetarightarrowinfty} f(theta v) = -infty$ and no minimizer exists.



          • Else, $b in Null(A)^{perp} = Col(A^T) = Col(A)$. Just find any vector $r$ such that $Ar=b$. Then:
            begin{align}
            frac{1}{2}(x-r)^TA(x-r) &= frac{1}{2}x^TAx - x^TAr + frac{1}{2}r^TAr \
            &= f(x) + frac{1}{2}r^TAr
            end{align}

            and this is clearly minimized at $x^*=r$, so
            $$ f(x^*) = f(r) = -frac{1}{2}r^TAr$$







          In other words, the problem reduces to Gaussian elimination: Find a vector $r$ such that $Ar=b$. If no such $r$ exists then no minimizer exists.






          share|cite|improve this answer























          • I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
            – Joe Lacey
            Nov 18 at 1:25













          up vote
          0
          down vote










          up vote
          0
          down vote









          I assume you are looking to minimize $f(x)$, and that $A$ is symmetric and positive semi-definite.




          • If $b notin Null(A)^{perp}$ then there is a vector $v in Null(A)$ such that $b^Tv neq 0$. Without loss of generality assume $b^Tv >0$ (else multiply $v$ by $-1$). Then for all real numbers $theta$ we get:
            $$ f(theta v) = 0 + (theta)(-b^Tv)$$
            so $lim_{thetarightarrowinfty} f(theta v) = -infty$ and no minimizer exists.



          • Else, $b in Null(A)^{perp} = Col(A^T) = Col(A)$. Just find any vector $r$ such that $Ar=b$. Then:
            begin{align}
            frac{1}{2}(x-r)^TA(x-r) &= frac{1}{2}x^TAx - x^TAr + frac{1}{2}r^TAr \
            &= f(x) + frac{1}{2}r^TAr
            end{align}

            and this is clearly minimized at $x^*=r$, so
            $$ f(x^*) = f(r) = -frac{1}{2}r^TAr$$







          In other words, the problem reduces to Gaussian elimination: Find a vector $r$ such that $Ar=b$. If no such $r$ exists then no minimizer exists.






          share|cite|improve this answer














          I assume you are looking to minimize $f(x)$, and that $A$ is symmetric and positive semi-definite.




          • If $b notin Null(A)^{perp}$ then there is a vector $v in Null(A)$ such that $b^Tv neq 0$. Without loss of generality assume $b^Tv >0$ (else multiply $v$ by $-1$). Then for all real numbers $theta$ we get:
            $$ f(theta v) = 0 + (theta)(-b^Tv)$$
            so $lim_{thetarightarrowinfty} f(theta v) = -infty$ and no minimizer exists.



          • Else, $b in Null(A)^{perp} = Col(A^T) = Col(A)$. Just find any vector $r$ such that $Ar=b$. Then:
            begin{align}
            frac{1}{2}(x-r)^TA(x-r) &= frac{1}{2}x^TAx - x^TAr + frac{1}{2}r^TAr \
            &= f(x) + frac{1}{2}r^TAr
            end{align}

            and this is clearly minimized at $x^*=r$, so
            $$ f(x^*) = f(r) = -frac{1}{2}r^TAr$$







          In other words, the problem reduces to Gaussian elimination: Find a vector $r$ such that $Ar=b$. If no such $r$ exists then no minimizer exists.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 17 at 22:54

























          answered Nov 17 at 22:49









          Michael

          13.1k11325




          13.1k11325












          • I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
            – Joe Lacey
            Nov 18 at 1:25


















          • I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
            – Joe Lacey
            Nov 18 at 1:25
















          I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
          – Joe Lacey
          Nov 18 at 1:25




          I completely agree with you for theoretical part. The problem is that we are facing a large ill-conditioned matrix. The goal is to find a good approximate solution. I should make the question clearer. Thanks
          – Joe Lacey
          Nov 18 at 1:25


















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