Does ADMM work for nonconvex optimization problems?












0












$begingroup$


I need to solve the following nonconvex optimization problem:
begin{equation}
begin{split}
min_{x,y}quad &f(x)+g(y)\
mathrm{s.t.}quad &Ax+By=b
end{split}
end{equation}

where $f$ is noncovex and $g$ is convex. A natural way is to use ADMM to solve this problem, which can be outlined as follows:



Define the augmented Lagrangian as
$$mathcal{L}_{beta}(x,y;omega)=f(x)+g(y)+w^{T}(Ax+By-b)+frac{beta}{2}||Ax+By-b||_2^2$$ then ADMM repeats as:



Step 1: $x^{k+1}inargmin_{x} mathcal{L}_{beta}(x,y^k;omega^k)$;



Step 2: $y^{k+1}inargmin_{y} mathcal{L}_{beta}(x^{k+1},y;omega^k)$;



Step 1: $omega^{k+1}=omega^{k}+beta(Ax^{k+1}+By^{k+1}-b)$;



As we know, ADMM works for convex optimization problrm with the guarantee of global convergence, but for this nonconvex problem, what's the convergence behavior?










share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    I need to solve the following nonconvex optimization problem:
    begin{equation}
    begin{split}
    min_{x,y}quad &f(x)+g(y)\
    mathrm{s.t.}quad &Ax+By=b
    end{split}
    end{equation}

    where $f$ is noncovex and $g$ is convex. A natural way is to use ADMM to solve this problem, which can be outlined as follows:



    Define the augmented Lagrangian as
    $$mathcal{L}_{beta}(x,y;omega)=f(x)+g(y)+w^{T}(Ax+By-b)+frac{beta}{2}||Ax+By-b||_2^2$$ then ADMM repeats as:



    Step 1: $x^{k+1}inargmin_{x} mathcal{L}_{beta}(x,y^k;omega^k)$;



    Step 2: $y^{k+1}inargmin_{y} mathcal{L}_{beta}(x^{k+1},y;omega^k)$;



    Step 1: $omega^{k+1}=omega^{k}+beta(Ax^{k+1}+By^{k+1}-b)$;



    As we know, ADMM works for convex optimization problrm with the guarantee of global convergence, but for this nonconvex problem, what's the convergence behavior?










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I need to solve the following nonconvex optimization problem:
      begin{equation}
      begin{split}
      min_{x,y}quad &f(x)+g(y)\
      mathrm{s.t.}quad &Ax+By=b
      end{split}
      end{equation}

      where $f$ is noncovex and $g$ is convex. A natural way is to use ADMM to solve this problem, which can be outlined as follows:



      Define the augmented Lagrangian as
      $$mathcal{L}_{beta}(x,y;omega)=f(x)+g(y)+w^{T}(Ax+By-b)+frac{beta}{2}||Ax+By-b||_2^2$$ then ADMM repeats as:



      Step 1: $x^{k+1}inargmin_{x} mathcal{L}_{beta}(x,y^k;omega^k)$;



      Step 2: $y^{k+1}inargmin_{y} mathcal{L}_{beta}(x^{k+1},y;omega^k)$;



      Step 1: $omega^{k+1}=omega^{k}+beta(Ax^{k+1}+By^{k+1}-b)$;



      As we know, ADMM works for convex optimization problrm with the guarantee of global convergence, but for this nonconvex problem, what's the convergence behavior?










      share|cite|improve this question











      $endgroup$




      I need to solve the following nonconvex optimization problem:
      begin{equation}
      begin{split}
      min_{x,y}quad &f(x)+g(y)\
      mathrm{s.t.}quad &Ax+By=b
      end{split}
      end{equation}

      where $f$ is noncovex and $g$ is convex. A natural way is to use ADMM to solve this problem, which can be outlined as follows:



      Define the augmented Lagrangian as
      $$mathcal{L}_{beta}(x,y;omega)=f(x)+g(y)+w^{T}(Ax+By-b)+frac{beta}{2}||Ax+By-b||_2^2$$ then ADMM repeats as:



      Step 1: $x^{k+1}inargmin_{x} mathcal{L}_{beta}(x,y^k;omega^k)$;



      Step 2: $y^{k+1}inargmin_{y} mathcal{L}_{beta}(x^{k+1},y;omega^k)$;



      Step 1: $omega^{k+1}=omega^{k}+beta(Ax^{k+1}+By^{k+1}-b)$;



      As we know, ADMM works for convex optimization problrm with the guarantee of global convergence, but for this nonconvex problem, what's the convergence behavior?







      optimization nonlinear-optimization numerical-optimization non-convex-optimization






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      edited Dec 16 '18 at 18:48









      Rodrigo de Azevedo

      13k41958




      13k41958










      asked Dec 15 '18 at 4:06









      ChenflChenfl

      215




      215






















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          In general, the convergence behavior can be arbitrarily bad. But, it all depends on the structure of $f(x)$. If you can find nice convex envelopes of the $f(x)$ you can get numerical bounds on the convergence. E.g., if $f(x)$ is bilinear, like $f(x)=x_1 x_2$. McCormick's relaxations provide envelopes https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes



          I would recommend finding convex envelopes to $f(x)$. Solving the relaxations like you would solve convex problems. Then evaluating the actual objective function at feasible points close to the solution of the enveloped functions.






          share|cite|improve this answer









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            $begingroup$

            In general, the convergence behavior can be arbitrarily bad. But, it all depends on the structure of $f(x)$. If you can find nice convex envelopes of the $f(x)$ you can get numerical bounds on the convergence. E.g., if $f(x)$ is bilinear, like $f(x)=x_1 x_2$. McCormick's relaxations provide envelopes https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes



            I would recommend finding convex envelopes to $f(x)$. Solving the relaxations like you would solve convex problems. Then evaluating the actual objective function at feasible points close to the solution of the enveloped functions.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              In general, the convergence behavior can be arbitrarily bad. But, it all depends on the structure of $f(x)$. If you can find nice convex envelopes of the $f(x)$ you can get numerical bounds on the convergence. E.g., if $f(x)$ is bilinear, like $f(x)=x_1 x_2$. McCormick's relaxations provide envelopes https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes



              I would recommend finding convex envelopes to $f(x)$. Solving the relaxations like you would solve convex problems. Then evaluating the actual objective function at feasible points close to the solution of the enveloped functions.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                In general, the convergence behavior can be arbitrarily bad. But, it all depends on the structure of $f(x)$. If you can find nice convex envelopes of the $f(x)$ you can get numerical bounds on the convergence. E.g., if $f(x)$ is bilinear, like $f(x)=x_1 x_2$. McCormick's relaxations provide envelopes https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes



                I would recommend finding convex envelopes to $f(x)$. Solving the relaxations like you would solve convex problems. Then evaluating the actual objective function at feasible points close to the solution of the enveloped functions.






                share|cite|improve this answer









                $endgroup$



                In general, the convergence behavior can be arbitrarily bad. But, it all depends on the structure of $f(x)$. If you can find nice convex envelopes of the $f(x)$ you can get numerical bounds on the convergence. E.g., if $f(x)$ is bilinear, like $f(x)=x_1 x_2$. McCormick's relaxations provide envelopes https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes



                I would recommend finding convex envelopes to $f(x)$. Solving the relaxations like you would solve convex problems. Then evaluating the actual objective function at feasible points close to the solution of the enveloped functions.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 16 '18 at 18:13









                skrskr

                17411




                17411






























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