Questions tagged [quasi-newton]
The quasi-newton tag has no usage guidance.
15
questions
2
votes
0answers
50 views
Does BFGS preserve the bandedness of the inverse hessian?
In the BFGS method we perform iterations by calculating an approximation $\boldsymbol{H}_k$ to the inverse Hessian $\boldsymbol{H}$ of the objective function. This method belongs to a family of ...
5
votes
1answer
108 views
Sensitivity of BFGS to the accuracy of the gradient
I am studying how to speed-up the BFGS method using quantum computing techniques.
I have used a method of speeding up the gradient of the function, but it sacrifices the precision value of the ...
1
vote
0answers
78 views
How to use Wolfe-Powell step-size control in quasi-Newton method?
I'm trying to find the minimum of a function using the quasi-Newton method with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm.
But I want to change the following implementation, so that:
1) ...
1
vote
1answer
72 views
Product of rank one updates as a low rank update for quasi newton/BFGS
I'm trying to improve the speed of the following iteration to calculate $s_k$:
$$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\...
3
votes
1answer
396 views
Computational complexity of Newton's method
the classical Newton's method for non-linear systems of equations is $x_{k+1} =x_k-J_F(x_n)^{-1} F(x_n)$. In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems $...
2
votes
0answers
203 views
How to prevent BFGS from getting stuck on astronomically large gradient?
I have implemented BFGS myself from scratch in order to solve minimization problems. Part of BFGS, as I understand it, is that the approximation to the Hessian is supposed to be positive definite, ...
3
votes
1answer
378 views
Linear constraints for L-BFGS-B
I know L-BFGS-B only supports simple box constrains of the form: $l_i \leq x_i \leq u_i$, where $l_i$ and $u_i$ are constants. For my specific optimization problem, I need to specify some simple ...
5
votes
0answers
85 views
Minimize interesting objective function with knowledge of gradient nonlinearity?
I plan on using a Quasi-Newton method (L-BFGS) to minimize a non-linear objective function.
$$ f: \mathbb{R}^n \rightarrow \mathbb{R}$$
The gradient is kind of interesting: as the values of the ...
3
votes
0answers
229 views
How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations?
I have to solve many times a nonlinear system of the form
$$f(x) = b^{(n)}$$
inside a loop.
The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
3
votes
2answers
695 views
Low-rank updates in BFGS
I have read this and other threads on this site on BFGS, but I still don't have a clear understanding of what it's meant by low-rank updates.
For example, I read the following in this book:
The ...
2
votes
1answer
62 views
Doubt regarding principled approach towards approximating the Hessian
In my optimization problem, the hessian has a structure such that it can be written as the sum of two matrices. Populating the first of the matrices is efficient. Populating the second one is ...
2
votes
0answers
128 views
Quasi-Newton Optimization with parallel function evaluation
I have a function of many variables (~200-2000) which I am optimizing with some success using L-BFGS. While the function is expensive to evaluate, the gradient can be computed with not much additional ...
2
votes
0answers
799 views
Weighted Frobenius norm in BFGS
In what sense is the weighted Frobenius norm "adimensional"/"scale-invariant" for any symmetric positive definite weight matrix $W$? If we plug in a positive diagonal matrix into $W$ wee see that $||A|...
1
vote
0answers
106 views
Quasi Newton taking very small steps
I have implemented a Quasi-Newton method based on the Hessian approximation. I am noticing that the algorithm takes too many iterations to converge, even though it does converge. What I am not able to ...
1
vote
1answer
103 views
The linear system in Quasi Newton method
I have implemented a Quasi Newton method for my problem, where I use the Hessian matrix approximation based approach. Hence, there is a linear system solve in every iteration. I solve the linear ...
