I conceptualize a following tri-level optimization problem: $$ \begin{aligned} & \min_{x_1,x_2,\theta} \ell(x_1,x_2) \\ s.t. \quad & x_2\in\arg\min_{x_2}\{\ell_2(x_2): \\ & \qquad g_2(x_1,x_2)\leq0 \\ & \qquad x_1 \in \arg\min_{x_1}\{\ell_1(x_1):g_1(x_1,x_2,\theta)\leq0\} \} \end{aligned} $$
My question is, can this problem be solved by a bi-level with two lower levels? E.g. $$ \begin{aligned} & \min_{x_1,x_2,\theta} \ell(x_1,x_2) \\ s.t. \quad & x_2\in\arg\min_{x_2}\{\ell_2(x_2): g_2(x_1,x_2)\leq0 \\ & x_1 \in \arg\min_{x_1}\{\ell_1(x_1):g_1(x_1,x_2,\theta)\leq0\} \} \end{aligned} $$
My initial thought is yes, at least the two formulations have the same optimal value, as the optimizers are feasible to each other. Any comments are much welcome.
