Convex relaxation in integer programming)

Computer vision problems are often attributed to the combination optimization problem. For such problems, the combination analysis method often cannot find the optimal solution within the polynomial time, therefore, we need a good approximation method to solve such problems. A reasonable method is to approach the integer programming problem through the convex relaxation method. The idea of this method is as follows:

First, let's look at the integer combination optimization problem. For the problem of finding the smallest vertex (red dot) in (Figure 1), the method of derivation is not available. The method of sorting is the NP problem, the optimal solution cannot be found within the polynomial time.

(Figure 1)

In this case, you can use the relaxation method to solve the problem. First, the range of the problem definition domain X is relaxed from the integer to the real value range, in addition, the target function is less than or equal to the original target function in the integer definition field, as shown in Figure 2:

(Figure 2)

In this case, the analysis method (such as gradient descent) can be used for processing, but there is still a problem that it may fall into the local minimum, the solution to local minimization is to make the target function convex in the definition field, so that it will not fall into the Local Minimization (like talking nonsense), as shown in Figure 3:

(Figure 3)

In this way, the approximate solution can be obtained directly through the analysis method and then rounded up.