Definitions and Conditions for optimality

This part follows closely Kelley 's Iterative Method for Optimization Kelley (1999). We start here with a series of definitions:

1.

$\bold{A}$ is positive definite if $\bold{x^TAx}\gt$ for all $\bold{x}\in\Re^N$

2.

$\bold{A}$ is spd if $\bold{A}$ is positive definite and symmetric

3.

$\bold{x}^* \in U$ ($U\subset\Re^N$) is a global minimizer if $f(\bold{x}^*)\leq f(\bold{x})$ for all $\bold{x} \in U$

The Euclidian norm is also defined as

$$\parallel \bold{x} \parallel = \sqrt{\sum_{i=1}^{N} (x_i)^2}.$$

Now, I give sufficient conditions that a minimizer $\bold{x}^*$ exists for a function f.