Optimum of coordinate-wise convex function












0












$begingroup$


Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:



$$
forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
$$



Can $f$ have a local optimum that is not also a global optimum?




I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).






optimization convex-analysis convex-optimization maxima-minima







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$endgroup$

















    0












    $begingroup$


    Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:



    $$
    forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
    $$



    Can $f$ have a local optimum that is not also a global optimum?




    I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).






    optimization convex-analysis convex-optimization maxima-minima







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:



      $$
      forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
      $$



      Can $f$ have a local optimum that is not also a global optimum?




      I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).






      optimization convex-analysis convex-optimization maxima-minima







      share|cite|improve this question









      $endgroup$




      Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:



      $$
      forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
      $$



      Can $f$ have a local optimum that is not also a global optimum?




      I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).






      optimization convex-analysis convex-optimization maxima-minima




      optimization convex-analysis convex-optimization maxima-minima






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 10 at 10:41









      PeterPeter

      356114




      356114






















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          $begingroup$

          The function
          $$
          f(x,y)
          =
          x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
          $$
          has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.






          share|cite|improve this answer









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            active

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            $begingroup$

            The function
            $$
            f(x,y)
            =
            x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
            $$
            has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              The function
              $$
              f(x,y)
              =
              x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
              $$
              has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                The function
                $$
                f(x,y)
                =
                x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
                $$
                has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.






                share|cite|improve this answer









                $endgroup$



                The function
                $$
                f(x,y)
                =
                x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
                $$
                has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 11 at 8:36









                gerwgerw

                19.4k11334




                19.4k11334






























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