# CHAPTER 8

# GRADIENT METHODS

# 8.1 Introduction

In this chapter we consider a class of search methods for real-valued functions on ^{n}. These methods use the gradient of the given function. In our discussion we use such terms as *level sets, normal vectors*, and *tangent vectors.* These notions were discussed in some detail in Part I.

Recall that a level set of a function *f* : ^{n} → is the set of points ** x** satisfying

**(**

*f***) =**

*x**c*for some constant

*c*. Thus, a point

*x*_{0}

^{n}is on the level set corresponding to level

*c*if

*f*(

*x*_{0}) =

*c*. In the case of functions of two real variables,

*f*:

^{2}→ , the notion of the level set is illustrated in Figure 8.1.

The gradient of *f* at *x*_{0}, denoted ∇*f*(*x*_{0}), if it is not a zero vector, is orthogonal to the tangent ...

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