# 在雙層編程中是否可能（或直接）定義許多次要問題？

Well, as @Matteo Fischetti has said in the comments to your question, your case of multiple "easy" secondary level problems is definitely more tractible than multiple levels. Even in a continuous linear case, you go one level higher in the polynomial hierarchy as you increase the number of levels [1].

But having multiple "easy" problems is not a complete triviality either. For example, $$Ax \leq b \\ 0\leq x \leq 1\\ y \geq 0\\ y_i \in \arg\min_y \{ y: y \geq -x_i; y \geq x_i -1 \} \} \quad i=1,\dots,n$$ This has $$n$$ simple bilevel constraints. Each program is trivial enough to have a closed form solution, $$y_i = \max \{x_i-1, -x_i\}$$! This along with the requirement $$y\geq 0$$ enforces that $$x\in\{0,1\}$$. In other words, we have re-written a binary program this way asking for one "trivial" bilevel constraint in the lifted space for each binary variable!

So you can regard having numerous trivial subproblems as having numerous binary variables.

That said, it is also fair to have numerous such bilevel constraints - they can be rewritten as a single one. In our case as $$(y_1, \dots, y_n) \in \arg\min_y \left \{ \sum_{i=1}^n y_i: y_i \geq -x_i; y_i \geq x_i -1 \quad i=1,\dots,n \right \}$$

References

[1] Jeroslow, R. G. (1985). The polynomial hierarchy and a simple model for competitive analysis. Mathematical Programming, 32(2), 146–164.