Figure 1

The SOCBLP problem with nonconvex feasible set. The optimal solution of the lower level problem of Example 3.1 is $y_1(x)-y_2(x)=6-0.5x$ if $2\le x\le 4$, and $y_1(x)-y_2(x)=8-x$ if $4\le x\le 6$, with $y_2(x)\ge 0$. The global optimal solutions are found at the points $D=\{(x,y)\in R_{+}\times R^2:x=6,y_1-y_2=2,y_2\ge 0\}$, which are depicted by the thick line in Figure 1, with an optimal function value of 12. It is shown by Figure 1 that, even in the simple case of functions, SOCBLP is a nonconvex and nondifferentiable optimization problem. Thus, it is possible that there exist local optimal solutions or stationary solutions for the SOCBLP. Furthermore, the optimal solutions of the lower level problem may be not unique in some cases.

The SOCBLP problem with nonconvex feasible set.