# Questions tagged [quasi-newton]

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### Difference between asymptotic and non-asymptotic convergence in optimization?

I am reading some optimization methods and I am facing some issues with two terms "asymptotic and non-asymptotic convergence". What is the difference between them?
34 views

### Interpretation of error between Hessian approximation and real Hessian - Quasi-Newton Method

$$||I- H_{k}^{BFGS}\nabla^{2}f(x_{k})||_{2}$$ , where $H_{k}$ is the inverse of hessian approximation at each iteration. I am given this expression to assess the error in Hessian approximation in ...
101 views

### General question related to BFGS

Based on my basic understanding of the BFGS method, the algorithm will iterate until the gradient norm is less than or equal to a set value called "gtol" in the case of Python. However, when ...
155 views

### When does L-BFGS outperform GD?

In practice, L-BFGS is frequently held comparably to other inexact QN methods, and it provides a middle ground of sorts between Hestenes–Stiefel CG and BFGS as memory goes from zero to infinity (...
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### Solution predictors for accelerating convergence in nonlinear FEM

I am looking for the details of commonly-used predictors for accelerating the convergence of iterations using Newton-Raphson scheme for nonlinear problems in FEM. I am looking specifically for static ...
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### 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 ...
154 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 ...
149 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) ...
78 views

I'm trying to improve the speed of the following iteration to calculate $s_k$: $$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\... 1answer 757 views ### Computational complexity of Newton's method the classical Newton's method for non-linear systems of equations is x_{k+1} =x_k-J_F(x_n)^{-1} F(x_n). In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems ... 0answers 354 views ### How to prevent BFGS from getting stuck on astronomically large gradient? I have implemented BFGS myself from scratch in order to solve minimization problems. Part of BFGS, as I understand it, is that the approximation to the Hessian is supposed to be positive definite, ... 1answer 590 views ### Linear constraints for L-BFGS-B I know L-BFGS-B only supports simple box constrains of the form: l_i \leq x_i \leq u_i, where l_i and u_i are constants. For my specific optimization problem, I need to specify some simple ... 0answers 88 views ### Minimize interesting objective function with knowledge of gradient nonlinearity? I plan on using a Quasi-Newton method (L-BFGS) to minimize a non-linear objective function.$$ f: \mathbb{R}^n \rightarrow \mathbb{R}$$The gradient is kind of interesting: as the values of the ... 0answers 298 views ### How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations? I have to solve many times a nonlinear system of the form$$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 ...
895 views

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 ...
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### 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 ...
148 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 ...
892 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|...