Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

Chapter 2
Optimization: an overview
2.1 What is optimization?
Intuitively, optimization refers to the class of problems that consists in choosing
the best among a set of alternatives.
Even in this simple, imprecise statement, one can identify the two fundamental
elements of an optimization problem: best, that conveys a choice of criterium
used to choose the solution; this is usually expressed by means of a function, that
should be minimized or maximized; alternatives, that refers to the set of possible
solutions that must be satisfied by any candidate solution. A simple example will
help us clarify these remarks.
Example 7 (Hotel-to-conference problem). Find the best street path to go from
the hotel where you are staying to the Convention Center. The alternatives here
consists of all of the streets (or parts of streets) that when joined provide a path
to go from your hotel to the Convention Center. It is clearly a finite set (provided
that we avoid paths containing loops). We have different choices for the term best:
• The street path that takes the smallest time;
• The shortest street path;
• The street path that has the best view of the landscape, etc.
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