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# 1.2 Some background on finite-dimensional optimization

Consider a function Let be some subset of , which could be the entire . We denote by the standard Euclidean norm on .

A point is a local minimum of over if there exists an such that for all satisfying we have

 (1.3)

In other words, is a local minimum if in some ball around it, does not attain a value smaller than . Note that this refers only to points in ; the behavior of outside is irrelevant, and in fact we could have taken the domain of to be rather than .

If the inequality in (1.3) is strict for , then we have a strict local minimum. If (1.3) holds for all , then the minimum is global over . By default, when we say a minimum" we mean a local minimum. Obviously, a minimum need not be unique unless it is both strict and global.

The notions of a (local, strict, global) maximum are defined similarly. If a point is either a maximum or a minimum, it is called an extremum. Observe that maxima of are minima of , so there is no need to develop separate results for both. We focus on the minima, i.e., we view as a cost function to be minimized (rather than a profit to be maximized).

Subsections

Next: 1.2.1 Unconstrained optimization Up: 1. Introduction Previous: 1.1 Optimal control problem   Contents   Index
Daniel 2010-12-20