An Introduction to Optimization, 4th Edition

CHAPTER 11

QUASI-NEWTON METHODS

11.1 Introduction

Newton’s method is one of the more successful algorithms for optimization. If it converges, it has a quadratic order of convergence. However, as pointed out before, for a general nonlinear objective function, convergence to a solution cannot be guaranteed from an arbitrary initial point x⁽⁰⁾ In general, if the initial point is not sufficiently close to the solution, then the algorithm may not possess the descent property for some k].

Recall that the idea behind Newton’s method is to locally approximate the function f being minimized, at every iteration, by a quadratic function. The minimizer for the quadratic approximation is used as the starting point for the next iteration. This leads to Newton’s recursive algorithm

We may try to guarantee that the algorithm has the descent property by modifying the original algorithm as follows:

where α_k is chosen to ensure that

For example, we may choose α_k = arg min_α≥0 f(x^(k) − αF(x^(k))⁻¹g^(k)) (see Theorem 9.2). We can then determine an appropriate value of α_k by performing a line search in the ...