Does ADMM work for nonconvex optimization problems?
$begingroup$
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?
optimization nonlinear-optimization numerical-optimization non-convex-optimization
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
asked Dec 15 '18 at 4:06
ChenflChenfl
214
$endgroup$
add a comment |
$begingroup$
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?
optimization nonlinear-optimization numerical-optimization non-convex-optimization
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
asked Dec 15 '18 at 4:06
ChenflChenfl
214
$endgroup$
add a comment |
$begingroup$
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?
optimization nonlinear-optimization numerical-optimization non-convex-optimization
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
asked Dec 15 '18 at 4:06
ChenflChenfl
214
$endgroup$
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
optimization nonlinear-optimization numerical-optimization non-convex-optimization
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
asked Dec 15 '18 at 4:06
ChenflChenfl
214
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
asked Dec 15 '18 at 4:06
ChenflChenfl
214
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
edited Dec 16 '18 at 18:48
Rodrigo de Azevedo
12.8k41855
12.8k41855
asked Dec 15 '18 at 4:06
ChenflChenfl
214
asked Dec 15 '18 at 4:06
ChenflChenfl
214
asked Dec 15 '18 at 4:06
ChenflChenfl
214
214
add a comment |
add a comment |
1 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.
answered Dec 16 '18 at 18:13
skrskr
17411
$endgroup$
add a comment |
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1 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.
answered Dec 16 '18 at 18:13
skrskr
17411
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 16 '18 at 18:13
skrskr
17411
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 16 '18 at 18:13
skrskr
17411
$endgroup$
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.
answered Dec 16 '18 at 18:13
skrskr
17411
answered Dec 16 '18 at 18:13
skrskr
17411
answered Dec 16 '18 at 18:13
skrskr
17411
answered Dec 16 '18 at 18:13
skrskr
17411
17411
add a comment |
add a comment |
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