1.3 Preview of infinite-dimensional optimization

Unlike in the case of
,
there does not exist a ``universal" function space. Many different choices
for
are possible, and specifying the desired space
is
part of the problem
formulation. Another issue is that in order
to define *local* minima of
over
, we
need to specify what it means for two functions in
to be close to each other.
Recall that in the definition of a local minimum in Section 1.2, a ball of radius
with respect to the standard Euclidean norm on
was
used to define the notion of closeness. In the present case we will again
employ
-balls, but we need to specify which norm we are going to use.
While in
all norms are equivalent (i.e., are within a constant multiple of one
another), in function spaces different choices of a norm lead to drastically
different notions of closeness. Thus, the first thing we need to do is become
more familiar with function spaces and norms on them.

- 1.3.1 Function spaces, norms, and local minima
- 1.3.2 First variation and first-order necessary condition
- 1.3.3 Second variation and second-order conditions
- 1.3.4 Global minima and convex problems