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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

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

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

 
next up previous print clean
Next: Theorem Up: Guitton: Huber solver Previous: Introduction
Stanford Exploration Project
4/27/2000