Figure 2

The SOCBLP problem with disconnected feasible set. The optimal solution of the lower level problem of Example 3.2 is $y_1(x)+y_2(x)=x$ if $2\le x\le 4$, and $y_1(x)+y_2(x)=8-x$ if $4\le x\le 6$, with $y_2(x)\le 0$. However, only for $x\in [2,3]\cup [5,6]$ the upper level constraint $y_1+y_2\le 3$ holds, and therefore the feasible set is not connected, which is depicted in Figure 2. Hence the global optimal solutions are found at $D=\{(x,y)\in R_{+}\times R^2:x=2,y_1+y_2=2,y_2\le 0\}$, which are the points on the thick line in Figure 2, with an optimal function value of 6. By Figure 2, we can see that the feasible set of the SOCBLP may be not connected especially when the upper level constraints depend on the lower level optimal solution.

The SOCBLP problem with disconnected feasible set.