Quasi-Newton Methods no-change Condition Requirement
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In standard quasi-newton methods for fixed point iteration, it looks there is two required conditions. The first one is secant condition:
$$J_{k+1} Delta x_{k} = Delta f_k $$
where $Delta f_k = f(x_{k+1}) - f(x_k)$, $Delta x_k = x_{k+1} - x_k$ and $J_{k+1}$ is the approximated Jocabian.
Furthermore, another common requirement is the following so-called no-change condition:
$$J_{k+1}q = J_k q$$
for each q such that:
$$q^{T}Delta x_{k}=0 $$
I couldn't understand the necessity of the second condition. Why do we avoid a new information along any direction q orthogonal to $Delta x_k$?
convex-optimization nonlinear-optimization newton-raphson
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
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add a comment |
$begingroup$
In standard quasi-newton methods for fixed point iteration, it looks there is two required conditions. The first one is secant condition:
$$J_{k+1} Delta x_{k} = Delta f_k $$
where $Delta f_k = f(x_{k+1}) - f(x_k)$, $Delta x_k = x_{k+1} - x_k$ and $J_{k+1}$ is the approximated Jocabian.
Furthermore, another common requirement is the following so-called no-change condition:
$$J_{k+1}q = J_k q$$
for each q such that:
$$q^{T}Delta x_{k}=0 $$
I couldn't understand the necessity of the second condition. Why do we avoid a new information along any direction q orthogonal to $Delta x_k$?
convex-optimization nonlinear-optimization newton-raphson
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
$endgroup$
add a comment |
$begingroup$
In standard quasi-newton methods for fixed point iteration, it looks there is two required conditions. The first one is secant condition:
$$J_{k+1} Delta x_{k} = Delta f_k $$
where $Delta f_k = f(x_{k+1}) - f(x_k)$, $Delta x_k = x_{k+1} - x_k$ and $J_{k+1}$ is the approximated Jocabian.
Furthermore, another common requirement is the following so-called no-change condition:
$$J_{k+1}q = J_k q$$
for each q such that:
$$q^{T}Delta x_{k}=0 $$
I couldn't understand the necessity of the second condition. Why do we avoid a new information along any direction q orthogonal to $Delta x_k$?
convex-optimization nonlinear-optimization newton-raphson
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
$endgroup$
In standard quasi-newton methods for fixed point iteration, it looks there is two required conditions. The first one is secant condition:
$$J_{k+1} Delta x_{k} = Delta f_k $$
where $Delta f_k = f(x_{k+1}) - f(x_k)$, $Delta x_k = x_{k+1} - x_k$ and $J_{k+1}$ is the approximated Jocabian.
Furthermore, another common requirement is the following so-called no-change condition:
$$J_{k+1}q = J_k q$$
for each q such that:
$$q^{T}Delta x_{k}=0 $$
I couldn't understand the necessity of the second condition. Why do we avoid a new information along any direction q orthogonal to $Delta x_k$?
convex-optimization nonlinear-optimization newton-raphson
convex-optimization nonlinear-optimization newton-raphson
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
asked Dec 3 '18 at 0:10
arincbulgurarincbulgur
11
11
add a comment |
add a comment |
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