126_Potential Constraint Set

In most problems, it's important to note that only some of the inequality constraints may be active at the minimum point. Moreover, the part of these activated constraints isn't known beforehand and must be determined during the problem-solving process. This post introduces the concept of potentially active constraints at the minimum point, a concept that can be integrated into numerical algorithms for constrained optimization for efficient computation.

 

 

The main reason for using a potential constraint strategy in an algorithm lies in the efficiency of the entire iterative process. This approach is especially effective for large-scale or complex programs where calculating the gradient of the constraints is costly. By employing a potential set strategy, only the gradients of the constraints within the set are calculated and used to define the subproblem for the search direction.

Although the original problem may have many constraints, only some may be included in the potential set, significantly reducing not only the number of gradient calculations but also the dimension of the subproblem defining the search direction. Thus computation time is significantly saved. Therefore, the potential set strategy is widely used in actual optimization programs.

 

 

In the $k$th iteration, the set of potential constraint indices can be defined as follows.

 

 

Setting ${x^{(k)}}$ to $(-4.5,-4.5)$ and $\varepsilon $ to $0.1$, and converting to normalized and standard form, the constrint equations can be represented as follows.

 

 

Here, since the second constraint lacks a constant, it is divided by 100. Calculating the constraints at the given point and identifying the active constraints results in the following.

 

 

From this, we can see that ${g_1}$ is active, ${g_4}$ and ${g_6}$ are violated, and ${g_2}$,${g_3}$, and ${g_5}$ are inactive. Therefore, the set of potential constraint indices is as follows.