Mixing
The Exponential Cone and Semi-definite programming
$begingroup$
I have a problem at the intersection of a range of topics: exponential programming, semi-definite programming and computer science, that I am having trouble finding a decent method for solving.
Take $A_iinmathbb{R}^{dtimes d}$ with $A_i = A^T_i$, $iinmathcal{I}$ and $mathcal{I}$ is a finite set of indices. $-infty < text{tr}(A_i)< 0, forall iinmathcal{I}$. We also have $b_iinmathbb{R}^+$.
We seek $X_iinmathbb{R}^{d times d}$ that solves the optimization problem
begin{align}
&min sum_i b_i e^{text{tr}(A_i X_i)} \
text{s.t.} & sum_i e^{text{tr}(X_i)} leq mathcal{C} \
& | X_i |^2_{Fr} leq alpha\
& X_i = X^T_i
end{align}
This can be solved using CVX, and thus satisfies Disciplined Convex Programming. The problem is that because it is semi-definite programming on the exponential cone, it requires an approximation of the cone, thus is quite slow. I have also used the large blunt object that is NLOPT which performs quickly, but this seems unsatisfactory given it doesn't really exploit the convex structure.
My question is: What other methods could I reasonably attack this problem with? In particular ones that might work in parallel (the real set $mathcal{I}$ is sufficiently large that the problem is distributed across many computers).
convex-optimization semidefinite-programming
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
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$endgroup$
add a comment |
$begingroup$
I have a problem at the intersection of a range of topics: exponential programming, semi-definite programming and computer science, that I am having trouble finding a decent method for solving.
Take $A_iinmathbb{R}^{dtimes d}$ with $A_i = A^T_i$, $iinmathcal{I}$ and $mathcal{I}$ is a finite set of indices. $-infty < text{tr}(A_i)< 0, forall iinmathcal{I}$. We also have $b_iinmathbb{R}^+$.
We seek $X_iinmathbb{R}^{d times d}$ that solves the optimization problem
begin{align}
&min sum_i b_i e^{text{tr}(A_i X_i)} \
text{s.t.} & sum_i e^{text{tr}(X_i)} leq mathcal{C} \
& | X_i |^2_{Fr} leq alpha\
& X_i = X^T_i
end{align}
This can be solved using CVX, and thus satisfies Disciplined Convex Programming. The problem is that because it is semi-definite programming on the exponential cone, it requires an approximation of the cone, thus is quite slow. I have also used the large blunt object that is NLOPT which performs quickly, but this seems unsatisfactory given it doesn't really exploit the convex structure.
My question is: What other methods could I reasonably attack this problem with? In particular ones that might work in parallel (the real set $mathcal{I}$ is sufficiently large that the problem is distributed across many computers).
convex-optimization semidefinite-programming
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
$endgroup$
1
$begingroup$
Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program.
$endgroup$
– Michael Grant
Jan 12 at 2:52
$begingroup$
And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable.
$endgroup$
– Michael Grant
Jan 12 at 2:54
add a comment |
$begingroup$
I have a problem at the intersection of a range of topics: exponential programming, semi-definite programming and computer science, that I am having trouble finding a decent method for solving.
Take $A_iinmathbb{R}^{dtimes d}$ with $A_i = A^T_i$, $iinmathcal{I}$ and $mathcal{I}$ is a finite set of indices. $-infty < text{tr}(A_i)< 0, forall iinmathcal{I}$. We also have $b_iinmathbb{R}^+$.
We seek $X_iinmathbb{R}^{d times d}$ that solves the optimization problem
begin{align}
&min sum_i b_i e^{text{tr}(A_i X_i)} \
text{s.t.} & sum_i e^{text{tr}(X_i)} leq mathcal{C} \
& | X_i |^2_{Fr} leq alpha\
& X_i = X^T_i
end{align}
This can be solved using CVX, and thus satisfies Disciplined Convex Programming. The problem is that because it is semi-definite programming on the exponential cone, it requires an approximation of the cone, thus is quite slow. I have also used the large blunt object that is NLOPT which performs quickly, but this seems unsatisfactory given it doesn't really exploit the convex structure.
My question is: What other methods could I reasonably attack this problem with? In particular ones that might work in parallel (the real set $mathcal{I}$ is sufficiently large that the problem is distributed across many computers).
convex-optimization semidefinite-programming
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
$endgroup$
I have a problem at the intersection of a range of topics: exponential programming, semi-definite programming and computer science, that I am having trouble finding a decent method for solving.
Take $A_iinmathbb{R}^{dtimes d}$ with $A_i = A^T_i$, $iinmathcal{I}$ and $mathcal{I}$ is a finite set of indices. $-infty < text{tr}(A_i)< 0, forall iinmathcal{I}$. We also have $b_iinmathbb{R}^+$.
We seek $X_iinmathbb{R}^{d times d}$ that solves the optimization problem
begin{align}
&min sum_i b_i e^{text{tr}(A_i X_i)} \
text{s.t.} & sum_i e^{text{tr}(X_i)} leq mathcal{C} \
& | X_i |^2_{Fr} leq alpha\
& X_i = X^T_i
end{align}
This can be solved using CVX, and thus satisfies Disciplined Convex Programming. The problem is that because it is semi-definite programming on the exponential cone, it requires an approximation of the cone, thus is quite slow. I have also used the large blunt object that is NLOPT which performs quickly, but this seems unsatisfactory given it doesn't really exploit the convex structure.
My question is: What other methods could I reasonably attack this problem with? In particular ones that might work in parallel (the real set $mathcal{I}$ is sufficiently large that the problem is distributed across many computers).
convex-optimization semidefinite-programming
convex-optimization semidefinite-programming
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
asked Jan 11 at 15:46
NeedsToKnowMoreMathsNeedsToKnowMoreMaths
728
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1
$begingroup$
Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program.
$endgroup$
– Michael Grant
Jan 12 at 2:52
$begingroup$
And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable.
$endgroup$
– Michael Grant
Jan 12 at 2:54
add a comment |
1
$begingroup$
Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program.
$endgroup$
– Michael Grant
Jan 12 at 2:52
$begingroup$
And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable.
$endgroup$
– Michael Grant
Jan 12 at 2:54
1
1
$begingroup$
Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program.
$endgroup$
– Michael Grant
Jan 12 at 2:52
$begingroup$
Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program.
$endgroup$
– Michael Grant
Jan 12 at 2:52
$begingroup$
And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable.
$endgroup$
– Michael Grant
Jan 12 at 2:54
$begingroup$
And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable.
$endgroup$
– Michael Grant
Jan 12 at 2:54
add a comment |
1 Answer
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There are solvers that can work directly with the exponential cone. Look at ECOS and the forthcoming Mosek version 9. If your problems are of a size where interior point methods are viable, one of those solvers might work for you.
What are the actual sizes of your problem instances?
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
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add a comment |
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$begingroup$
There are solvers that can work directly with the exponential cone. Look at ECOS and the forthcoming Mosek version 9. If your problems are of a size where interior point methods are viable, one of those solvers might work for you.
What are the actual sizes of your problem instances?
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
$endgroup$
add a comment |
$begingroup$
There are solvers that can work directly with the exponential cone. Look at ECOS and the forthcoming Mosek version 9. If your problems are of a size where interior point methods are viable, one of those solvers might work for you.
What are the actual sizes of your problem instances?
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
$endgroup$
add a comment |
$begingroup$
There are solvers that can work directly with the exponential cone. Look at ECOS and the forthcoming Mosek version 9. If your problems are of a size where interior point methods are viable, one of those solvers might work for you.
What are the actual sizes of your problem instances?
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
$endgroup$
There are solvers that can work directly with the exponential cone. Look at ECOS and the forthcoming Mosek version 9. If your problems are of a size where interior point methods are viable, one of those solvers might work for you.
What are the actual sizes of your problem instances?
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
edited Jan 11 at 21:46
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
answered Jan 11 at 21:29
Brian BorchersBrian Borchers
6,07111219
6,07111219
add a comment |
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$begingroup$
Your terminology is a bit off. This isn't semidefinite programming. Semidefinite programs can only have linear matrix inequalities for nonlinearities, and this has none of those. This is simply a smooth nonlinear program.
$endgroup$
– Michael Grant
Jan 12 at 2:52
$begingroup$
And I see nothing whatsoever wrong with using NLOPT, if its performance and accuracy are acceptable.
$endgroup$
– Michael Grant
Jan 12 at 2:54