How to minimize $sum_i,j |M[i,j] -(v_1(i) - v_2(j))^2|$

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Say I have an $n$ by $n$ matrix $M$ of positive real numbers. For two vectors $v_1$ and $v_2$ of length $n$ each, I say that the cost of the vectors is:



$$sum_i,j |M[i,j] -(v_1(i) - v_2(j))^2|$$



How can you choose $v_1$ and $v_2$ to minimize this cost? I can't work out if there is a straightforward formulation of this optimization problem.




optimization




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    Say I have an $n$ by $n$ matrix $M$ of positive real numbers. For two vectors $v_1$ and $v_2$ of length $n$ each, I say that the cost of the vectors is:



    $$sum_i,j |M[i,j] -(v_1(i) - v_2(j))^2|$$



    How can you choose $v_1$ and $v_2$ to minimize this cost? I can't work out if there is a straightforward formulation of this optimization problem.




    optimization




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      up vote
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      down vote

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      up vote
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      down vote

      favorite











      Say I have an $n$ by $n$ matrix $M$ of positive real numbers. For two vectors $v_1$ and $v_2$ of length $n$ each, I say that the cost of the vectors is:



      $$sum_i,j |M[i,j] -(v_1(i) - v_2(j))^2|$$



      How can you choose $v_1$ and $v_2$ to minimize this cost? I can't work out if there is a straightforward formulation of this optimization problem.




      optimization




      share|cite|improve this question











      Say I have an $n$ by $n$ matrix $M$ of positive real numbers. For two vectors $v_1$ and $v_2$ of length $n$ each, I say that the cost of the vectors is:



      $$sum_i,j |M[i,j] -(v_1(i) - v_2(j))^2|$$



      How can you choose $v_1$ and $v_2$ to minimize this cost? I can't work out if there is a straightforward formulation of this optimization problem.





      optimization





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      share|cite|improve this question









      asked Jul 31 at 16:33









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