Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

2.5. VARIATIONAL OPTIMIZATION PROBLEMS 27
Example 10 (The Traveling Salesman Problem). This need is well illustrated by
the Traveling Salesman Problem. Given n towns, one wishes to find the minimum
length route that starts in a given town, goes through each one of the others and
ends at the starting town.
The combinatorial structure of the problem is quite simple. Each possible route
corresponds to one of the (n − 1)! circular permutations of the n towns. Thus, it
suffices to enumerate these permutations, evaluate their lengths, and choose the
optimal route.
This, however, becomes unpractical even for moderate values of n. For instance,
for n = 50 there are approximatelly 10 60 permutations to examine. Even
if 1 billion of them were evaluated per second, examining all would require about
10 51 seconds, or 10 43 years! However, there are techniques that allow solving this
problem, in practice, even for larger values of n.
2.5 Variational optimization problems
An optimization problem is called a variational problem when its solution set S
is an infinite dimensional subset of a space of functions.
Among the most important examples we could mention the path and surface
problems. The problems consist in finding the “best path” (“best surface”) satisfying
some conditions which define the solution set.
Typical examples of variational problems are:
Example 11 (Geodesic problem). Find the path of minimum length joining two
points p 1 and p 2 of a given surface.
Example 12 (Minimal surface problem). Find the surface of minimum area for a
given boundary curve.
Variational problems are studied in more detaill in Chapter 4.
2.6 Other classifications
Other forms of classifying optimization problems are based on characteristics
which can be exploited in order to devise strategies for the solution.