# Difference between revisions of "Quasi-Newton methods"

Author: Jianmin Su (ChemE 6800 Fall 2020)

Steward: Allen Yang, Fengqi You

Quasi-Newton Methods are a kind of methods used to solve nonlinear optimization problems. They are based on Newton’s method yet can be an alternative of Newton’s method when the objective function is not twice-differentiable, which means the Hessian matrix is unavailable, or it is too expensive to calculate the Hessian matrix and its inverse.

## Introduction

The first quasi-Newton algorithm was developed by W.C. Davidon in the mid1950s and it turned out to be a milestone in nonlinear optimization problems. He was trying to solve a long optimization calculation but he failed to get the result with the original method due to the low performances of computers at that time, thus he managed to build the quasi-Newton method to solve it. Later then, Fletcher and Powell proved that the new algorithm was more efficient and more reliable than the other existing methods.

During the following years, numerous variants were proposed, include Broyden's method (1965), the SR1 formula (Davidon 1959, Broyden 1967), the DFP method (Davidon, 1959; Fletcher and Powell, 1963), and the BFGS method (Broyden, 1969; Fletcher, 1970; Goldfarb, 1970; Shanno, 1970).

In optimization problems, Newton's method uses first and second derivatives, gradient and the Hessian in multivariate scenarios, to find the optimal point, it is applied to a twice-differentiable function $f$ to find the roots of the first derivative (solutions to $f'(x)=0$ ), also known as the stationary points of $f$ .

The iteration of Newton's method is usually written as: $x_{k+1}=x_{k}-H^{-1}\cdot \bigtriangledown f(x_{k})$ , where$k$ is the iteration number, $H$ is the Hessian matrix and $H=[\bigtriangledown ^{2}f(x_{k})]$ Iteraton would stop when it satisfies the convergence criteria like ${df \over dx}=0,||\bigtriangledown f(x)||<\epsilon {\text{ or }}|f(x_{k+1})-f(x_{k})|<\epsilon$ Quasi-Newton methods are similar to Newton's method but with one key idea that is different, they don't calculate the Hessian matrix, they introduce a matrix $B$ to estimate the Hessian matrix instead so that they can avoid the time-consuming calculations of Hessian matrix and its inverse. And there are many variants of quasi-Newton methods that simply depend on the exact methods they use in the estimation of the Hessian matrix.

## Theory and Algorithm

formulas that will use

the objective function at current iterate $x_{k}$ B_k

$m_{k}(p)=f_{k}+\bigtriangledown f_{k}^{T}p+{\frac {1}{2}}p^{T}B_{k}p$ the objective function at iterate $x_{k}+1$ to update B_k+1

$m_{k+1}(p)=f_{k+1}+\bigtriangledown f_{k+1}^{T}p+{\frac {1}{2}}p^{T}B_{k+1}p$ $B_{k+1}\alpha _{k}p_{k}=\bigtriangledown f_{k+1}-\bigtriangledown f_{k}$ Define $s_{k}=x_{k+1}-x_{k}$ , $y_{k}=\bigtriangledown f_{k+1}-\bigtriangledown f_{k}$ So that $B_{k+1}s_{k}=y_{k}$ , which is secant equation, can maintain $s_{k}^{T}B_{k+1}s_{k}>0$ , along with Wolfe conditions to maintain is minimizing the objective function.

To further prove $B_{k+1}$ is positive definite , converge

$B_{k+1}={\underset {B}{min}}||B-B_{k}||$ Using different norms will lead to different methods that used to update $B_{k}$ $||A||_{W}=||W^{\frac {1}{2}}AW^{\frac {1}{2}}||_{F}$ , the right is Frobenius norm, W is the mean of Hessian matrix

$B_{k+1}=(I-\rho y_{k}s_{k}^{T})B_{k}(I-\rho s_{k}y_{k}^{T})+\rho y_{k}y_{k}^{T}$ $\rho ={\frac {1}{y_{k}^{T}s_{k}}}$ Set $H_{k}=B_{k}^{-1}$ with Sherman-Morrison formaula, we can get H_{k+1}=H_k+\frac{s_k s_k^T}{s_k^T y_k}-\frac{H_k y_k y_k^T H_k}{y_k^T H_k y_k}

In the DFP method, we use $B_{k}$ to estimate the inverse of Hessian matrix

In the BFGS method, we use $B_{k}$ to estimate the Hessian matrix

$B_{k}H_{k}$ ### DFP Algorithm

1. Given the starting point $x_{0}$ ; convergence tolerance $\epsilon ,\epsilon >0$ ; the initial estimation of inverse Hessian matrix $D_{0}=I$ ; $k=0$ .
2. Compute the search direction $d_{k}=-D_{k}\cdot g_{k}$ .
3. Compute the step length $\lambda _{k}$ with $\lambda =\arg {\underset {\lambda \in \mathbb {R} }{min}}f(x_{k}+\lambda d_{k}),$ , and then set$s_{k}={\lambda }_{k}d_{k}$ , then $x_{k+1}=x_{k}+s_{k}$ 4. If $||g_{k+1}||<\epsilon$ , then end of the iteration, otherwise continue step5.
5. Computing $y_{k}=g_{k+1}-g_{k}$ .
6. Update the $D_{k+1}$ with$D_{k+1}=D_{k}+{\frac {s_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}}-{\frac {D_{k}y_{k}y_{k}^{T}D_{k}}{y_{k}^{T}D_{k}y_{k}}}$ 7. Update $k$ with $k=k+1$ and go back to step2.

### BFGS Algorithm

1. Given the starting point $x_{0}$ ; convergence tolerance $\epsilon ,\epsilon >0$ ; the initial estimation of Hessian matrix $B_{0}=I$ ; $k=0$ .
2. Compute the search direction $d_{k}=-B_{k}^{-1}\cdot g_{k}$ .
3. Compute the step length $\lambda _{k}$ with $\lambda =\arg {\underset {\lambda \in \mathbb {R} }{min}}f(x_{k}+\lambda d_{k}),$ , and then set$s_{k}={\lambda }_{k}d_{k}$ , then $x_{k+1}=x_{k}+s_{k}$ 4. If $||g_{k+1}||<\epsilon$ , then end of the iteration, otherwise continue step5.
5. Computing $y_{k}=g_{k+1}-g_{k}$ .
6. Update the $B_{k+1}$ with$B_{k+1}=B_{k}+{\frac {y_{k}y_{k}^{T}}{y_{k}^{T}s_{k}}}-{\frac {B_{k}s_{k}s_{k}^{T}B_{k}}{s_{k}^{T}B_{k}s_{k}}}$ 7. Update $k$ with $k=k+1$ and go back to step2.

## Numerical Example

{\begin{aligned}f(x_{1},x_{2})&=x_{1}^{2}+{\frac {1}{2}}x_{2}^{2}+3\end{aligned}} 