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# 1.3 Preview of infinite-dimensional optimization

In Section 1.2 we considered the problem of minimizing a function . Now, instead of we want to allow a general vector space , and in fact we are interested in the case when this vector space is infinite-dimensional. Specifically, will itself be a space of functions. Let us denote a generic function in by , reserving the letter for the argument of . (This will typically be a scalar, and has no relation with from the previous section.) The function to be minimized is a real-valued function on , which we now denote by . Since is a function on a space of functions, it is called a functional. To summarize, we are minimizing a functional .

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.

Subsections

Next: 1.3.1 Function spaces, norms, Up: 1. Introduction Previous: 1.2.2.2 Second-order conditions   Contents   Index
Daniel 2010-12-20