Convergence properties of most minimization algorithms are analyzed through
their application to convex * quadratic functions*. Such functions
can be written in the form of
equation (3),
where **A** is a positive-definite matrix.
We refer to this convex quadratic function throughout
this chapter by .
For such a function, the unique * global* minimum
satisfies the linear system
Since general functions can be
approximated by a quadratic convex function in the neighborhood of their
local minima, the convergence properties obtained for convex quadratic
functions are usually applied locally to general functions.
However, such generalizations do not guarantee good behavior in
practice on complex, large-scale functions.