I. Introduction
In this paper, we consider a class of bilevel optimization problems as follows \begin{align*} &\mathrm{minimize}\qquad h(x) \qquad\qquad \qquad \qquad \qquad \qquad (P_{f}^{h})\\ &\text{subject to}\qquad x\in X^{\ast}\triangleq\arg\min_{\mathrm{v}\in X}f(y). \end{align*} where denote the lower and upper level objective functions, respectively, and is a constraint set. This is called the selection problem ([9], [23]) as we are selecting among optimal solutions of a lower level problem, one that minimizes the objective function h. In particular, we consider the case where the lower level objective function is given as , where is the th component function for .