Neural network to solve nonlinear constrained optimization problem












0












$begingroup$


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.










share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















0












$begingroup$


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.










share|cite|improve this question











$endgroup$












  • $begingroup$
    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














0












0








0


0



$begingroup$


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.










share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 14 '18 at 18:35







Clifford

















asked Dec 10 '18 at 19:35









CliffordClifford

11




11












  • $begingroup$
    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












  • $begingroup$
    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










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