Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

2.3. DISCRETE OPTIMIZATION PROBLEMS 25
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Figure 2.2: Discrete solution sets.
It is important to remark that some types of discrete problems min{f(x); x ∈ S}
with solution set S, may be solved more easily by embedding S into a continuous
domain S ′ , solving the new, continuous, optimization problem min{f(x); x ∈
S ′ }, and obtaining the solution of the original problem from the solution in the
continuous domain. The example below illustrates this approach.
Example 9. Consider the discrete optimization problem max{f(x) = 3x −
2x 2 ; x ∈ Z}. In order to solve this problem we consider the similiar problem
on R, that is, max{f(x) = 3x − 2x 2 ; x ∈ R}.
The solution of the continuous problem is simple. In fact, since f(x) is a
quadratic function with negative second derivative, it has a unique maximum point
given by f ′ (x) = 0, that is, 3 − 4x = 0. Therefore, the solution is m = 3/4, and
the maximum value is 9/8. Now we obtain the solution of the discrete problem
from this solution.
Note that the solution obtained for S = R is not a solution for the discrete
case (because m ∉ Z). But it is possible to compute the solution in the discrete
case from the solution of the continuous problem. For this, we just need to choose
the integer number which is closest to m. It is possible to prove that this provides
indeed the solution. In fact, since f is an increasing function in the interval
[−∞, m], and decreasing in the interval [m, ∞] and its graph is symmetrical with