Neural network to solve nonlinear constrained optimization problem
$begingroup$
I'm trying to build a neural network to solve this optimization problem
minimize $f(x)$
s.t. $h(x)=0$
where $x =(x_1, x_2, dots, x_n)^T in R^n$, $f: R^n rightarrow R$ and $h: R^n rightarrow R^m$ are given functions and $m le n$. $f$ and $h$ are assumed to be twice continuous differentiable.
The idea is presented in this paper (IEEE abstract)
and consists in creating a neural network whose equilibrium point satisfies the necessary conditions of optimality.
Based on the Lagrange multiplier theory, the neural network will be governed by:
$frac{dx}{dt} = - nabla_x L(x, lambda) $
$frac{dlambda}{dt} = nabla_{lambda} L(x, lambda) $
where $L: R^{m+n} rightarrow R$ is the Lagrange function defined by: $L(x, lambda) = f(x) + lambda^T h(x)$, $lambda in R^m$
- I do not see how we could use the last equations to create a neural
network? - For a neural network, the output $(x^*, lambda^*)$ "must" be known,
also, how to implement forward propagation and backpropagation? - Could someone help me understand how to create such a network or, if
possible, provide a Matlab or Python code to implement the network?
Many thanks.
nonlinear-optimization neural-networks
asked Dec 10 '18 at 19:35
CliffordClifford
162
$endgroup$
add a comment |
$begingroup$
I'm trying to build a neural network to solve this optimization problem
minimize $f(x)$
s.t. $h(x)=0$
where $x =(x_1, x_2, dots, x_n)^T in R^n$, $f: R^n rightarrow R$ and $h: R^n rightarrow R^m$ are given functions and $m le n$. $f$ and $h$ are assumed to be twice continuous differentiable.
The idea is presented in this paper (IEEE abstract)
and consists in creating a neural network whose equilibrium point satisfies the necessary conditions of optimality.
Based on the Lagrange multiplier theory, the neural network will be governed by:
$frac{dx}{dt} = - nabla_x L(x, lambda) $
$frac{dlambda}{dt} = nabla_{lambda} L(x, lambda) $
where $L: R^{m+n} rightarrow R$ is the Lagrange function defined by: $L(x, lambda) = f(x) + lambda^T h(x)$, $lambda in R^m$
- I do not see how we could use the last equations to create a neural
network? - For a neural network, the output $(x^*, lambda^*)$ "must" be known,
also, how to implement forward propagation and backpropagation? - Could someone help me understand how to create such a network or, if
possible, provide a Matlab or Python code to implement the network?
Many thanks.
nonlinear-optimization neural-networks
asked Dec 10 '18 at 19:35
CliffordClifford
162
$endgroup$
$begingroup$
You can do these kind of things, but I don't know how efficient it would be..
$endgroup$
– mathreadler
Dec 14 '18 at 18:50
$begingroup$
Any idea to start with or any web ressource ?
$endgroup$
– Clifford
Dec 15 '18 at 18:00
add a comment |
$begingroup$
I'm trying to build a neural network to solve this optimization problem
minimize $f(x)$
s.t. $h(x)=0$
where $x =(x_1, x_2, dots, x_n)^T in R^n$, $f: R^n rightarrow R$ and $h: R^n rightarrow R^m$ are given functions and $m le n$. $f$ and $h$ are assumed to be twice continuous differentiable.
The idea is presented in this paper (IEEE abstract)
and consists in creating a neural network whose equilibrium point satisfies the necessary conditions of optimality.
Based on the Lagrange multiplier theory, the neural network will be governed by:
$frac{dx}{dt} = - nabla_x L(x, lambda) $
$frac{dlambda}{dt} = nabla_{lambda} L(x, lambda) $
where $L: R^{m+n} rightarrow R$ is the Lagrange function defined by: $L(x, lambda) = f(x) + lambda^T h(x)$, $lambda in R^m$
- I do not see how we could use the last equations to create a neural
network? - For a neural network, the output $(x^*, lambda^*)$ "must" be known,
also, how to implement forward propagation and backpropagation? - Could someone help me understand how to create such a network or, if
possible, provide a Matlab or Python code to implement the network?
Many thanks.
nonlinear-optimization neural-networks
asked Dec 10 '18 at 19:35
CliffordClifford
162
$endgroup$
I'm trying to build a neural network to solve this optimization problem
minimize $f(x)$
s.t. $h(x)=0$
where $x =(x_1, x_2, dots, x_n)^T in R^n$, $f: R^n rightarrow R$ and $h: R^n rightarrow R^m$ are given functions and $m le n$. $f$ and $h$ are assumed to be twice continuous differentiable.
The idea is presented in this paper (IEEE abstract)
and consists in creating a neural network whose equilibrium point satisfies the necessary conditions of optimality.
Based on the Lagrange multiplier theory, the neural network will be governed by:
$frac{dx}{dt} = - nabla_x L(x, lambda) $
$frac{dlambda}{dt} = nabla_{lambda} L(x, lambda) $
where $L: R^{m+n} rightarrow R$ is the Lagrange function defined by: $L(x, lambda) = f(x) + lambda^T h(x)$, $lambda in R^m$
- I do not see how we could use the last equations to create a neural
network? - For a neural network, the output $(x^*, lambda^*)$ "must" be known,
also, how to implement forward propagation and backpropagation? - Could someone help me understand how to create such a network or, if
possible, provide a Matlab or Python code to implement the network?
Many thanks.
nonlinear-optimization neural-networks
nonlinear-optimization neural-networks
asked Dec 10 '18 at 19:35
CliffordClifford
162
asked Dec 10 '18 at 19:35
CliffordClifford
162
edited Dec 14 '18 at 18:35
Clifford
asked Dec 10 '18 at 19:35
CliffordClifford
162
asked Dec 10 '18 at 19:35
CliffordClifford
162
asked Dec 10 '18 at 19:35
CliffordClifford
162
162
$begingroup$
You can do these kind of things, but I don't know how efficient it would be..
$endgroup$
– mathreadler
Dec 14 '18 at 18:50
$begingroup$
Any idea to start with or any web ressource ?
$endgroup$
– Clifford
Dec 15 '18 at 18:00
add a comment |
$begingroup$
You can do these kind of things, but I don't know how efficient it would be..
$endgroup$
– mathreadler
Dec 14 '18 at 18:50
$begingroup$
Any idea to start with or any web ressource ?
$endgroup$
– Clifford
Dec 15 '18 at 18:00
$begingroup$
You can do these kind of things, but I don't know how efficient it would be..
$endgroup$
– mathreadler
Dec 14 '18 at 18:50
$begingroup$
You can do these kind of things, but I don't know how efficient it would be..
$endgroup$
– mathreadler
Dec 14 '18 at 18:50
$begingroup$
Any idea to start with or any web ressource ?
$endgroup$
– Clifford
Dec 15 '18 at 18:00
$begingroup$
Any idea to start with or any web ressource ?
$endgroup$
– Clifford
Dec 15 '18 at 18:00
add a comment |
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$begingroup$
You can do these kind of things, but I don't know how efficient it would be..
$endgroup$
– mathreadler
Dec 14 '18 at 18:50
$begingroup$
Any idea to start with or any web ressource ?
$endgroup$
– Clifford
Dec 15 '18 at 18:00