Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

2.2. CONTINUOUS OPTIMIZATION PROBLEMS 23
• the description (definition) of the solution set S;
• the properties of the objective function f.
The most important classification is the one based on the nature of the solution
set S, that leads us to classifying optimization problems into four classes:
continuous, discrete, combinatorial and variational.
A point x of a topological space S is an accumulation point, if, and only if, for
any open ball B x , with x ∈ B x , there exists an element y ∈ S such that y ∈ B x .
A point x ∈ S which is not an accumulation point is called an isolated point of S.
Thus, x is isolated if there exists an open ball B x , such that x ∈ B x and no other
point of S belongs to B x .
A topological space is discrete if it contains no accumulation points. It is continuous
when all of its points are accumulation points.
Example 8 (Discrete and continuous sets). Any finite subset of an euclidean
space is obviously discrete. The set R n itself is continuous. The set Z n =
{(i 1 , . . . , i n ); i j ∈ Z} is discrete. The set {0} ∪ {1/n; n ∈ Z} is not discrete because
0 is an accumulation point (no matter how small we take a ball with center
at 0, it will contain some element 1/n for some large n); this set is not continuous
either, because with the exception of 0 all of the points are isolated. On the other
hand, the set {1/n; n ∈ Z} (without the 0) is discrete.
2.2 Continuous optimization problems
An optimization problem is called continuous when the solution set S is a continuous
subset of R n .
The most common cases occur when S is a differentiable m-dimensional surface
of R n (with or without boundary), or S ⊆ R n is a region of R n :
• Maximize the function f(x, y) = 3x + 4y + 1 on the set S = {(x, y) ∈
R 2 ; x 2 + y 2 = 1}. The solution set S is the unit circle S 1 , which is a curve
of R 2 (i.e. a one-dimensional “surface”, see Figure 2.1(a)).
• Maximize the function f(x, y) = 3x + 4y + 1 on the set S = {(x, y) ∈
R 2 ; x 2 + y 2 ≤ 1}. The solution set is the disk of R 2 , a two-dimensional
surface with boundary (see Figure 2.1(b)).