# 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 ${\displaystyle f}$ to find the roots of the first derivative (solutions to ${\displaystyle f'(x)=0}$), also known as the stationary points of ${\displaystyle f}$.

The iteration of Newton's method is usually written as: ${\displaystyle x_{k+1}=x_{k}-H^{-1}\cdot \bigtriangledown f(x_{k})}$, where ${\displaystyle k}$ is the iteration number, ${\displaystyle H}$ is the Hessian matrix and ${\displaystyle H=[\bigtriangledown ^{2}f(x_{k})]}$

Iteraton would stop when it satisfies the convergence criteria like ${\displaystyle {df \over dx}=0,||\bigtriangledown f(x)||<\epsilon {\text{ or }}|f(x_{k+1})-f(x_{k})|<\epsilon }$

Though we can solve an optimization problem quickly with Newton's method, it has two obvious disadvantages:

1. The objective function must be twice-differentiable and the Hessian matrix must be positive definite.
2. The calculation is costly because it requires to compute the Jacobian matrix, Hessian matrix and its inverse, which is time-consuming when dealing with a large-scale optimization problem.

However, we can use Quasi-Newton methods to avoid these two disadvantages.·

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 ${\displaystyle B}$ 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

To illustrate the basic idea behind quasi-Newton methods, we start with building a quadratic model of the objective function at the current iterate ${\displaystyle x_{k}}$:

${\displaystyle m_{k}(p)=f_{k}+\bigtriangledown f_{k}^{T}p+{\frac {1}{2}}p^{T}B_{k}p}$ (1.1), where ${\displaystyle B_{k}}$ is an ${\displaystyle n\times n}$ symmetric positive definite matrix that will be updated at every iteration.

The minimizer of this convex quadratic model is:

${\displaystyle p_{k}=-B_{k}^{-1}\bigtriangledown f_{k}}$ (1.2), which is also used as the search direction.

Then the new iterate could be written as: ${\displaystyle x_{k+1}=x_{k}+\alpha _{k}p_{k}}$ (1.3),

where ${\displaystyle \alpha _{k}}$ is the step length that should satisfy the Wolfe conditions. The iteration is similar to Newton's method, but we use the approximate Hessian ${\displaystyle B_{k}}$ instead of the true Hessian.

To maintain the curve information we got from the previous iteration in ${\displaystyle B_{k+1}}$, we generate a new iterate ${\displaystyle x_{k+1}}$ and new quadratic modelto in the form of:

${\displaystyle m_{k+1}(p)=f_{k+1}+\bigtriangledown f_{k+1}^{T}p+{\frac {1}{2}}p^{T}B_{k+1}p}$ (1.4).

To construct the relationship between 1.1 and 1.4, we require that in 1.1 at ${\displaystyle p=0}$ the function value and gradient match ${\displaystyle f_{k}}$ and ${\displaystyle \bigtriangledown f_{k}}$, and the gradient of ${\displaystyle m_{k+1}}$should match the gradient of the objective function at the latest two iterates ${\displaystyle x_{k}}$and ${\displaystyle x_{k+1}}$, then we can get:

${\displaystyle \bigtriangledown m_{k+1}(-\alpha _{k}p_{k})=\bigtriangledown f_{k+1}-\alpha _{k}B_{k+1}p_{k}=\bigtriangledown f_{k}}$ (1.5)

and with some arrangements:

${\displaystyle B_{k+1}\alpha _{k}p_{k}=\bigtriangledown f_{k+1}-\bigtriangledown f_{k}}$ (1.6)

Define ${\displaystyle s_{k}=x_{k+1}-x_{k}}$, ${\displaystyle y_{k}=\bigtriangledown f_{k+1}-\bigtriangledown f_{k}}$ (1.7)

So that 1.6 becomes: ${\displaystyle B_{k+1}s_{k}=y_{k}}$, which is the secant equation.

To make sure ${\displaystyle B_{k+1}}$ is still a symmetric positive definite matrix, we need ${\displaystyle s_{k}^{T}s_{k}>0}$.

To further preserve properties of ${\displaystyle B_{k+1}}$ and determine ${\displaystyle B_{k+1}}$ uniquely,

${\displaystyle B_{k+1}={\underset {B}{min}}||B-B_{k}||}$

Using different norms will lead to different methods that used to update ${\displaystyle B_{k}}$

${\displaystyle ||A||_{W}=||W^{\frac {1}{2}}AW^{\frac {1}{2}}||_{F}}$, the right is Frobenius norm, W is the mean of Hessian matrix

${\displaystyle B_{k+1}=(I-\rho y_{k}s_{k}^{T})B_{k}(I-\rho s_{k}y_{k}^{T})+\rho y_{k}y_{k}^{T}}$

${\displaystyle \rho ={\frac {1}{y_{k}^{T}s_{k}}}}$

Set ${\displaystyle H_{k}=B_{k}^{-1}}$ with Sherman-Morrison formaula, we can get ${\displaystyle H_{k+1}=H_{k}+{\frac {s_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}}-{\frac {H_{k}y_{k}y_{k}^{T}H_{k}}{y_{k}^{T}H_{k}y_{k}}}}$

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

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

${\displaystyle B_{k}H_{k}}$

### DFP Algorithm

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

### BFGS Algorithm

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

## Numerical Example

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